Gelfand–Phillips Type Properties of Locally Convex Spaces

Abstract

We let $1\le p\le q\le \infty .$ Being motivated by the classical notions of the Gelfand–Phillips property and the (coarse) Gelfand–Phillips property of order p of Banach spaces, we introduce and study different types of the Gelfand–Phillips property of order $(p,q)$ (the $G{P}_{(p,q)}$ property) and the coarse Gelfand–Phillips property of order p in the realm of all locally convex spaces. We compare these classes and show that they are stable under taking direct product, direct sums and closed subspaces. It is shown that any locally convex space is a quotient space of a locally convex space with the $G{P}_{(p,q)}$ property. Characterizations of locally convex spaces with the introduced Gelfand–Phillips type properties are given.

1. Introduction

All locally convex spaces are assumed to be Hausdorff and over the field $\mathbb{F}$ of real or complex numbers. We denote by $E^{\prime }$ the topological dual of a locally convex space (lcs for short) E. The topological dual space $E^{\prime }$ of E endowed with the weak* topology $\sigma (E^{\prime },E)$ and the strong topology $\beta (E^{\prime },E)$ is denoted by $E_{w^{*}}^{\prime }$ and $E_{\beta }^{\prime }$, respectively. For a bounded subset $B\subseteq E$ and a functional $\chi \in E^{\prime }$, we put

$${\parallel \chi \parallel }_B:=sup\left\{\left|\chi \right(x\left)\right|:x\in B\cup \left\{0\right\}\right\}.$$

If E is a Banach space, we denote by $B_E$ the closed unit ball of E.

Definition 1.

A bounded subset B of a Banach space E is called limited if each weak* null sequence ${\left\{{\chi }_n\right\}}_{n\in \omega }$ in $E^{\prime }$ converges to zero uniformly on B, that is, ${lim}_{n\to \infty }{\parallel {\chi }_n\parallel }_B=0$.

Let us recall the classical notion of Gelfand–Phillips spaces.

Definition 2.

A Banach space E is said to have the Gelfand–Phillips property or is a Gelfand–Phillips space if every limited set in E is relatively compact.

In [1], Gelfand proved that every separable Banach space is Gelfand–Phillips. On the other hand, Phillips [2] showed that the non-separable Banach space ${\ell }_{\infty }$ is not Gelfand–Phillips.

Following Bourgain and Diestel [3], a bounded linear operator T from a Banach space L into E is called limited if $T\left(B_L\right)$ is a limited subset of E. In [4], Drewnowski noticed the next characterization of Banach spaces with the Gelfand–Phillips property.

Theorem 1

([4]). For a Banach space E, the following assertions are equivalent:

(i)

E is Gelfand–Phillips;

(ii)

every limited weakly null sequence in E is norm null;

(iii)

every limited operator with range in E is compact.

This characterization of Gelfand–Phillips Banach spaces plays a crucial role in many arguments for establishing the Gelfand–Phillips property in Banach spaces. The Gelfand–Phillips property was intensively studied in particular in [4,5,6,7,8,9]. It follows from results of Schlumprecht [7,8] that the Gelfand–Phillips property is not a three space property (see also Theorem 6.8.h in [10]). In our recent paper [11], we offer several new characterizations of Gelfand–Phillips Banach spaces.

Another direction for studying the Gelfand–Phillips property is to characterize Gelfand–Phillips spaces that belong to some important classes of Banach spaces. In the next proposition, whose short proof is given in Corollary 2.2 of [11], we provide some of the most important and general results in this direction (for all relevant definitions used in what follows, see Section 2).

Theorem 2.

A Banach space E is Gelfand–Phillips if one of the following conditions holds:

(i)

([11], Cor. 2.2) the closed unit ball $B_{E^{\prime }}$ of the dual space $E^{\prime }$ endowed with the weak* topology is selectively sequentially pseudocompact;

(ii)

([1]) E is separable;

(iii)

([5], Prop. 2) E is separably weak*-extensible;

(iv)

(cf. [4], Th. 2.2 and [8], Prop. 2) the space $E_{w^{*}}^{\prime }$ is selectively sequentially pseudocompact at some E-norming set $S\subseteq E^{\prime }$;

(v)

([6], Th. 4.1) $E=C\left(K\right)$ for some compact selectively sequentially pseudocompact space K.

The notion of a limited set in locally convex spaces was introduced by Lindström and Schlumprecht in [12] and independently by Banakh and Gabriyelyan in [13]. Since limited sets in the sense of [12] are defined using equicontinuity, to distinguish both notions, we call them in [13] by $\mathcal{E}$-limited sets:

Definition 3.

A subset A of a locally convex space E is called

(i)

$\mathcal{E}$-limited if $\parallel {\chi }_n{\parallel }_A\to 0$ for every equicontinuous weak* null sequence ${\left\{{\chi }_n\right\}}_{n\in \omega }$ in $E^{\prime }$ ([12]);

(ii)

limited if $\parallel {\chi }_n{\parallel }_A\to 0$ for every weak* null sequence ${\left\{{\chi }_n\right\}}_{n\in \omega }$ in $E^{\prime }$ ([13]).

It is clear that if E is a $c_0$-barrelled space, then A is limited if and only if it is $\mathcal{E}$-limited, and for Banach spaces, both these notions of limited sets coincide.

Lindström and Schlumprecht [12] proposed the following extention of the Gelfand–Phillips property to the class of all locally convex spaces.

Definition 4

([12]). A locally convex space E is called an $\mathcal{E}$-Gelfand–Phillips space or it has the $\mathcal{E}$-Gelfand–Phillips property if every $\mathcal{E}$-limited set in E is precompact.

Also, in Definition 4, we use the notation of “$\mathcal{E}$-Gelfand–Phillips property” instead of the “Gelfand–Phillips property” as in [12] to emphasize the equicontinuity of weak* null sequences in $E^{\prime }$ (as we noticed above, this extra condition can be omitted for $c_0$-barrelled spaces but not in general) and the fact that this notion can be naturally extended from the case of Banach spaces to the general case not in a unique way.

Fréchet spaces with the $\mathcal{E}$-Gelfand–Phillips property were thoroughly studied by Alonso in [14]. The following operator characterization of $\mathcal{E}$-Gelfand–Phillips spaces was obtained by Ruess in [15].

Theorem 3

([15]). A locally convex space E is an $\mathcal{E}$-Gelfand–Phillips space if and only if for each locally convex (equivalently, Fréchet or Banach) space L, a subset H of the ϵ-product $EϵL$ of E and L is precompact if and only if

(i)

$\left\{T\right(\chi ):T\in H\}$ is precompact in L for all $\chi \in E^{\prime }$;

(ii)

for any equicontinuous, weak* null sequence ${\left\{{\chi }_n\right\}}_{n\in \omega }$ in $E^{\prime }$, it follows that $T\left({\chi }_n\right)\to 0$ in L uniformly over all $T\in H$.

A generalization of the Gelfand–Phillips property to the class of all locally convex spaces can be performed not in a unique way as in Definition 4. Motivated by the notion of the Josefson–Nissenzweig property, the next notion was proposed in [13]. We say that a subset A of a locally convex space (lcs for short) E is barrel-bounded or barrel-precompact if A is bounded or, respectively, precompact in the space $\left(E,\beta (E,E^{\prime })\right)$ (all relevant definitions are given in Section 2).

Definition 5

([13]). A locally convex space E is said to have the b-Gelfand–Phillips property or else E is a b-Gelfand–Phillips space if every limited barrel-bounded subset of E is barrel-precompact.

It is easy to see that a Banach space E has the Gelfand–Phillips property if and only if it is b-Gelfand–Phillips in the sense of Definition 5 (where the prefix “b-” is added to emphasize the “barreled” nature of the notion); and if E is a barrelled space, then E has the b-Gelfand–Phillips property if and only if it is an $\mathcal{E}$-Gelfand–Phillips space or a Gelfand–Phillips space in the sense of Lindström and Schlumprecht [12].

We let E and H be locally convex spaces. We denote by $\mathcal{L}(E,H)$ the family of all operators (=continuous linear maps) from E to H. We let $p\in [1,\infty ]$. We recall that sequence ${\left\{x_n\right\}}_{n\in \omega }$ in E is called weakly p-summable if for every $\chi \in E^{\prime }$ it follows that $\left(\langle \chi ,x_n\rangle \right)\in {\ell }_p$ if $p\in [1,\infty )$ and $\left(\langle \chi ,x_n\rangle \right)\in c_0$ if $p=\infty$. The families ${\ell }_p^w\left(E\right)$ and $c_0^w\left(E\right)$ of all weakly p-summable sequences in E are vector spaces which admit natural locally convex vector topologies such that they are complete if so is E; for details, see Section 19.4 in [16] or Section 4 in [17].

Unifying the notion of the unconditional convergent (u.c.) operator and the notion of completely continuous operators (i.e., they transform weakly null sequences into norm null), Castillo and Sánchez selected in [18] the class of p-convergent operators. An operator $T:X\to Y$ between Banach spaces X and Y is called p-convergent if it transforms weakly p-summable sequences into norm null sequences. Using this notion, they introduced and studied Banach spaces with the Dunford–Pettis property of order p ($DP{P}_p$ for short) for every $p\in [1,\infty ]$. A Banach space X is said to have the $DP{P}_p$ if every weakly compact operator from X into a Banach space Y is p-convergent.

The influential article of Castillo and Sánchez [18] inspired an intensive study of p-versions of numerous geometrical properties of Banach spaces. In particular, the following p-versions of limitedness were introduced by Karn and Sinha [19] and Galindo and Miranda [20].

Definition 6.

We let $p\in [1,\infty ]$ and let X be a Banach space. A bounded subset A of X is called

(i)

a p-limited set if

$$(\underset{a\in A}{sup}\left|\langle {\chi }_n,a\rangle \right|)\in {\ell }_p\left(or(\underset{a\in A}{sup}\left|\langle {\chi }_n,a\rangle \right|)\in c_{0}ifp=\infty \right)$$

for every $\left({\chi }_n\right)\in {\ell }_p^w\left(X^{*},\sigma (X^{*},X)\right)$ (or $\left({\chi }_n\right)\in c_0^w\left(X^{*},\sigma (X^{*},X)\right)$ if $p=\infty$) ([19]);

(ii)

a coarse p-limited set if for every $T\in \mathcal{L}(E,{\ell }_p)$(or $T\in \mathcal{L}(E,c_0)$ if $p=\infty$), the set $T\left(A\right)$ is relatively compact ([20]).

Now, it is natural to define the (coarse) p-version of the Gelfand–Phillips property of Banach spaces as follows.

Definition 7.

We let $p\in [1,\infty ]$. A Banach space X is said to have

(i)

the p-Gelfand–Phillips property (the $G{P}_p$ property) if every limited weakly p-summable sequence in X is norm null ([21]);

(ii)

the coarse p-Gelfand–Phillips property (the coarse $G{P}_p$ property) if every coarse p-limited set in E is relatively compact ([20]).

Taking into account Theorem 1, it is easy to see that a Banach space X has the $G{P}_{\infty }$ property if and only if it has the $GP$ property. Numerous results concerning the Gelfand–Phillips type properties were obtained by Deghhani at al. [22], Fourie and Zeekoei [23], Ghenciu [24,25,26], and Galindo and Miranda [20].

The classes of limited, $\mathcal{E}$-limited, p-limited, and coarse p-limited sets can be defined in any locally convex space as follows.

Definition 8

([27]). We let $1\le p\le q\le \infty$. A non-empty subset A of a locally convex space E is called

(i)

a $(p,q)$-limited set (resp., $(p,q)$-$\mathcal{E}$-limited set) if

$$\left(\parallel {\chi }_n{\parallel }_A\right)\in {\ell }_{q}ifq<\infty ,or{\parallel {\chi }_n\parallel }_A\to 0ifq=\infty ,$$

for every (resp., equicontinuous) weak* p-summable sequence ${\left\{{\chi }_n\right\}}_{n\in \omega }$ in $E^{\prime }$;

(ii)

a coarse p-limited set if for every $T\in \mathcal{L}(E,{\ell }_p)$ (or $T\in \mathcal{L}(E,c_0)$ if $p=\infty$), the set $T\left(A\right)$ is relatively compact.

We denote by ${\mathsf{L}}_{(p,q)}\left(E\right)$ and ${\mathsf{EL}}_{(p,q)}\left(E\right)$ the family of all $(p,q)$-limited subsets and all $(p,q)$-$\mathcal{E}$-limited subsets of E, respectively. $(p,p)$-($\mathcal{E}$-)limited sets and $(\infty ,\infty )$-($\mathcal{E}$-)limited sets are called simply p-($\mathcal{E}$-)limited sets and ($\mathcal{E}$-)limited sets, respectively. The family of all coarse p-limited sets is denoted by ${\mathsf{CL}}_p\left(E\right)$.

Taking into consideration that the classes of precompact, sequentially precompact, relatively sequentially compact and relatively compact subsets of a locally convex space are distinct in general, one can naturally define the following types of the Gelfand–Phillips property generalizing the corresponding notions from Definitions 2, 4, 5, and 7.

Definition 9.

We let $1\le p\le q\le \infty$. A locally convex space E is said to have

(i)

a $(p,q)$-Gelfand–Phillips property (a precompact $(p,q)$-Gelfand–Phillips property, a sequential $(p,q)$-Gelfand–Phillips property, or a sequentially precompact $(p,q)$-Gelfand–Phillips property) if every $(p,q)$-limited set in E is relatively compact (resp., precompact, relatively sequentially compact or sequentially precompact); in short, the $G{P}_{(p,q)}$ property (resp., the $prG{P}_{(p,q)}$ property, the $sG{P}_{(p,q)}$ property, or the $spG{P}_{(p,q)}$ property);

(ii)

a $(p,q)$-equicontinuous Gelfand–Phillips property (a precompact $(p,q)$-equicontinuous Gelfand–Phillips property, a sequential $(p,q)$-equicontinuous Gelfand–Phillips property, or a sequentially precompact $(p,q)$-equicontinuous Gelfand–Phillips property) if every $(p,q)$-$\mathcal{E}$-limited set in E is relatively compact (resp., precompact, relatively sequentially compact or sequentially precompact); in short, the $EG{P}_{(p,q)}$ property (resp., the $prEG{P}_{(p,q)}$ property, the $sEG{P}_{(p,q)}$ property, or the $spEG{P}_{(p,q)}$ property);

(iii)

a b-$(p,q)$-Gelfand–Phillips property (a b-$(p,q)$-equicontinuous Gelfand–Phillips property) if every $(p,q)$-limited (resp., $(p,q)$-$\mathcal{E}$-limited) set in E is barrel-precompact; in short, the b-$G{P}_{(p,q)}$ property or the b-$EG{P}_{(p,q)}$ property, respectively;

(iv)

a coarse p-Gelfand–Phillips property (a coarse precompact p-Gelfand–Phillips property, a coarse sequential p-Gelfand–Phillips property, or a coarse sequentially precompact p-Gelfand–Phillips property) if every coarse p-limited set in E is relatively compact (resp., precompact, relatively sequentially compact, or sequentially precompact); in short, the coarse $G{P}_p$ property (resp., the coarse $prG{P}_p$ property, the coarse $sG{P}_p$ property, or the coarse $spG{P}_p$ property).

In the case $q=p$, we write only one subscript p, and if $q=p=\infty$, the subscripts are omitted (so we say that E has the $G{P}_p$ property or the $GP$ property, etc., respectively).

The main purpose of the article is to study locally convex spaces with Gelfand–Phillips type properties introduced in Definition 9. Now, we describe the content of the article.

In Section 2, we fix basic notions and prove some necessary results used in the article. It is a well known result due to Odell and Stegall (see [28], p. 377) that any limited set in a Banach space is weakly sequentially precompact. In Theorem 4, we essentially generalize this result using the same idea. By the celebrated Rosenthal ${\ell }_1$-theorem, any bounded sequence in a Banach space E has a weakly Cauchy subsequence if and only if E has no isomorphic copy of ${\ell }_1$. The remarkable result of Ruess [29] shows that an analogous statement holds true for much wider classes of locally convex spaces. These facts motivate us to introduce the weak Cauchy subsequence property of order p ($wCS{P}_p$ for short) for any $p\in [1,\infty ]$; see Definition 14. In Proposition 2, we prove that if $1<p<\infty$, then ${\ell }_p$ has $wCS{P}_p$ if and only if $p\ge 2$. We also show that the class of locally convex spaces with $wCS{P}_p$ is stable under taking countable products and arbitrary direct sums; see Proposition 3.

In Section 3, we select natural relationships between Gelfand–Phillips type properties introduced in Definition 9; see Lemmas 2, 3 and 5 and Propositions 6 and 6. As a corollary, we show in Proposition 4 that if E is an angelic, complete and barrelled space (for example, E is a strict $\left(LF\right)$-space), then all Gelfand–Phillips type properties from (i)–(iii) of Definition 9 coincide. If the space E has some of Schur type properties, then $V^{*}$ type properties of Pełczyński’ imply corresponding Gelfand–Phillips type properties; see Proposition 7. It is easy to see (Lemma 2) that the Gelfand–Phillips type properties of order $(1,\infty )$ are the strongest ones in the sense that if E has for example the $spG{P}_{(1,\infty )}$ property, then it has the $spG{P}_{(p,q)}$ property for all $1\le p\le q\le \infty$. Therefore, one can expect that spaces with $spG{P}_{(1,\infty )}$ property have some additional properties. This is indeed so; in Proposition 11, we show that if E is a sequentially complete barrelled locally convex space whose bounded subsets are weakly sequentially precompact, then E has the $spG{P}_{(1,\infty )}$ property if and only if it has the Schur property. In Proposition 8, we show that the classes of locally convex spaces with Gelfand–Phillips type properties are stable under taking direct products and direct sums, and Proposition 9 states heredity properties of Gelfand–Phillips type properties. In Theorem 5, we characterize spaces $C_p\left(X\right)$ which have one of the equicontinuous $GP$ properties from (ii) of Definition 9. In Theorem 6, we show that any lcs is a quotient space of an lcs with the $G{P}_{(p,q)}$ property; consequently, the class of locally convex spaces with the $G{P}_{(p,q)}$ property is sufficiently rich.

Characterizations of locally convex spaces with Gelfand–Phillips type properties are given in Section 4. In Theorem 10, we obtain an operator characterization of locally convex spaces with the sequential $G{P}_{(p,q)}$ property and the sequentially precompact $G{P}_{(p,q)}$ property; our results generalize a characterization of Banach spaces with the Gelfand–Phillips property of order p obtained by Ghenciu in Theorem 15 of [24]. In Theorem 11, we characterize locally convex spaces with the sequentially precompact $G{P}_{(p,q)}$ property in an important class of locally convex spaces which includes all strict $\left(LF\right)$-spaces. In Theorem 12, we essentially generalize the equivalence (i)⇔(ii) in Theorem 1. A natural direct extension of the Gelfand–Phillips property for Banach spaces is the precompact $G{P}_{(p,\infty )}$ property with $p\in [1,\infty ]$ (so that the precompact Gelfand–Phillips property and the precompact $G{P}_{(1,\infty )}$ property are two extreme cases). Locally convex spaces with the precompact $G{P}_{(p,\infty )}$ property are characterized in Theorem 13. Sufficient conditions to have the precompact $G{P}_{(p,\infty )}$ property are given in Theorem 14 and Corollary 2. Banach spaces with the coarse p-Gelfand–Phillips property are characterized by Galindo and Miranda in Proposition 13 of [20]. In Theorem 15, we generalize their result to all locally convex spaces.

Recall that a Banach space E has the Gelfand–Phillips property if and only if every limited weakly null sequence in E is norm null; see Theorem 1 (or the more general Theorem 12 below). This characterization motivates us to introduce the following type of Gelfand–Phillips property which is studied in Section 5.

Definition 10.

We let $p,q,q^{\prime }\in [1,\infty ]$, $q^{\prime }\le q$. A locally convex space $(E,\tau )$ is said to have the p-Gelfand–Phillips sequentially compact property of order $(q^{\prime },q)$ (p-$GPsc{P}_{(q^{\prime },q)}$ for short) if every weakly p-summable sequence in E which is also a $(q^{\prime },q)$-limited set is τ-null. If $q^{\prime }=q$, $q^{\prime }=q=\infty$, $p=\infty$, or $p=q^{\prime }=q=\infty$, we say simply that E has the p-$GPsc{P}_q$, the p-$GPscP$, $GPsc{P}_{(q^{\prime },q)}$, or the $GPscP$, respectively.

We characterize locally convex spaces E with the p-$GPsc{P}_{(q^{\prime },q)}$ in Theorem 16. This theorem extends and generalizes characterizations of Banach spaces with the Gelfand–Phillips property obtained in Theorem 2.2 of [30] and Corollary 13(ii) of [25]. Spaces with the $GPsc{P}_{(q,\infty )}$ are characterized in Theorem 18.

In Section 6, we introduce strong versions of $V^{*}$ type properties of Pełczyński, which usually are stronger than the corresponding Gelfand–Phillips type properties. These new classes of locally convex spaces have nice stability properties; see Propositions 19 and 20. Locally convex spaces with the strong (resp., sequentially) precompact $V_p^{*}$ property are characterized in Theorem 19.

2. Preliminaries Results

We start with some necessary definitions and notations used in the article. We set $\omega :=\{0,1,2,\dots \}$. The closed unit ball of the field $\mathbb{F}$ is denoted by $\mathbb{D}$. All topological spaces are assumed to be Tychonoff (= completely regular and $T_1$). The closure of a subset A of a topological space X is denoted by $\overline{A}$. The space $C\left(X\right)$ of all continuous functions on X endowed with the pointwise topology is denoted by $C_p\left(X\right)$. A Tychonoff space X is called Fréchet–Urysohn if for any cluster point $a\in X$ of a subset $A\subseteq X$ there is a sequence ${\left\{a_n\right\}}_{n\in \omega }\subseteq A$ which converges to a. A Tychonoff space X is called an angelic space if (1) every relatively countably compact subset of X is relatively compact and (2) any compact subspace of X is Fréchet–Urysohn. We note that any subspace of an angelic space is angelic, and a subset A of an angelic space X is compact if and only if it is countably compact if and only if A is sequentially compact; see Lemma 0.3 of [31].

We let E be a locally convex space. The span of subset A of E and its closure are denoted by $E_A:=\mathsf{span}\left(A\right)$ and $\overline{\mathsf{span}}\left(A\right)$, respectively. We denote by ${\mathcal{N}}_0\left(E\right)$ (resp., ${\mathcal{N}}_0^c\left(E\right)$) the family of all (resp., closed absolutely convex) neighborhoods of zero of E. The family of all bounded subsets of E is denoted by $\mathsf{Bo}\left(E\right)$. The value of $\chi \in E^{\prime }$ on $x\in E$ is denoted by $\langle \chi ,x\rangle$ or $\chi \left(x\right)$. Sequence ${\left\{x_n\right\}}_{n\in \omega }$ in E is said to be Cauchy if for every $U\in {\mathcal{N}}_0\left(E\right)$ there is $N\in \omega$ such that $x_n-x_m\in U$ for all $n,m\ge N$. If E is a normed space, $B_E$ denotes the closed unit ball of E. The family of all operators from E to an lcs L is denoted by $\mathcal{L}(E,L)$.

For an lcs E, we denote by $E_w$ and $E_{\beta }$ the space E endowed with the weak topology $\sigma (E,E^{\prime })$ and with the strong topology $\beta (E,E^{\prime })$, respectively. The topological dual space $E^{\prime }$ of E endowed with weak* topology $\sigma (E^{\prime },E)$ or with the strong topology $\beta (E^{\prime },E)$ is denoted by $E_{w^{*}}^{\prime }$ or $E_{\beta }^{\prime }$, respectively. The polar of a subset A of E is denoted by $A^{\circ }:=\{\chi \in E^{\prime }{:\parallel \chi \parallel }_A\le 1\}.$ A subset B of $E^{\prime }$ is equicontinuous if $B\subseteq U^{\circ }$ for some $U\in {\mathcal{N}}_0\left(E\right)$.

A subset A of a locally convex space E is called

• precompact if for every $U\in {\mathcal{N}}_0\left(E\right)$ there is a finite set $F\subseteq E$ such that $A\subseteq F+U$;
• sequentially precompact if every sequence in A has a Cauchy subsequence;
• weakly (sequentially) compact if A is (sequentially) compact in $E_w$;
• relatively weakly compact if its weak closure ${\overline{A}}^{\sigma (E,E^{\prime })}$ is compact in $E_w$;
• relatively weakly sequentially compact if each sequence in A has a subsequence weakly converging to a point of E;
• weakly sequentially precompact if each sequence in A has a weakly Cauchy subsequence.

Note that each sequentially precompact subset of E is precompact, but the converse is not true in general, see Lemma 2.2 of [17]. We use repeatedly the next lemma; see Lemma 4.4 in [32].

Lemma 1.

We let τ and $\mathcal{T}$ be two locally convex vector topologies on a vector space E such that $\tau \subseteq \mathcal{T}$. If $S={\left\{x_n\right\}}_{n\in \omega }$ is τ-null and $\mathcal{T}$-precompact, then S is $\mathcal{T}$-null. Consequently, if S is weakly $\mathcal{T}$-null and $\mathcal{T}$-precompact, then S is $\mathcal{T}$-null.

In what follows, we actively use the following classical completeness type properties and weak barrelledness conditions. A locally convex space E is

• quasi-complete if each closed bounded subset of E is complete;
• sequentially complete if each Cauchy sequence in E converges;
• locally complete if the closed absolutely convex hull of a null sequence in E is compact;
• (quasi)barrelled if every $\sigma (E^{\prime },E)$-bounded (resp., $\beta (E^{\prime },E)$-bounded) subset of $E^{\prime }$ is equicontinuous;
• $c_0$-(quasi)barrelled if every $\sigma (E^{\prime },E)$-null (resp., $\beta (E^{\prime },E)$-null) sequence is equicontinuous.

It is well known that $C_p\left(X\right)$ is quasibarrelled for every Tychonoff space X.

We denote by ${⨁}_{i\in I}{E}_i$ and ${\prod }_{i\in I}{E}_i$ the locally convex direct sum and the topological product of a non-empty family ${\left\{E_i\right\}}_{i\in I}$ of locally convex spaces, respectively. If $0\ne \mathbf{x}=\left(x_i\right)\in {⨁}_{i\in I}{E}_i$, then the set $\mathrm{supp}\left(\mathbf{x}\right):=\{i\in I:x_i\ne 0\}$ is called the support of $\mathbf{x}$. The support of a subset A, $\left\{0\right\}⊊A$, of ${⨁}_{i\in I}{E}_i$ is the set $\mathrm{supp}\left(A\right):={\bigcup }_{a\in A}\mathrm{supp}\left(a\right)$. We also consider elements $\mathbf{x}=\left(x_i\right)\in {\prod }_{i\in I}{E}_i$ as functions on I and write $\mathbf{x}\left(i\right):=x_i$.

We denote by ${\mathsf{ind}}_{n\in \omega }{E}_n$ the inductive limit of a (reduced) inductive sequence ${\left\{(E_n,{\tau }_n)\right\}}_{n\in \omega }$ of locally convex spaces. If in addition ${\tau }_m{\upharpoonright }_{E_n}={\tau }_n$ for all $n,m\in \omega$ with $n\le m$, the inductive limit ${\mathsf{ind}}_{n\in \omega }{E}_n$ is called strict and is denoted by ${\underrightarrow{s-\mathsf{ind}}}_n{E}_n$. In the partial case when all spaces $E_n$ are Fréchet, the strict inductive limit is called a strict $\left(LF\right)$-space.

Below, we recall some of the basic classes of compact type operators.

Definition 11.

We let E and L be locally convex spaces. An operator $T\in \mathcal{L}(E,L)$ is called compact (resp., sequentially compact, precompact, sequentially precompact, weakly compact, weakly sequentially compact, weakly sequentially precompact, bounded) if there is $U\in {\mathcal{N}}_0\left(E\right)$ such that $T\left(U\right)$ is a relatively compact (relatively sequentially compact, precompact, sequentially precompact, relatively weakly compact, relatively weakly sequentially compact, weakly sequentially precompact or bounded) subset of E.

We let $p\in [1,\infty ]$. Then, $p^{*}$ is defined to be the unique element of $[1,\infty ]$ which satisfies $\frac{1}{p}+\frac{1}{p^{*}}=1$. For $p\in [1,\infty )$, the space ${\ell }_{p^{*}}$ is the dual space of ${\ell }_p$. We denote by ${\left\{e_n\right\}}_{n\in \omega }$ the canonical basis of ${\ell }_p$ if $1\le p<\infty$ or the canonical basis of $c_0$ if $p=\infty$. The canonical basis of ${\ell }_{p^{*}}$ is denoted by ${\left\{e_n^{*}\right\}}_{n\in \omega }$. In what follows, we usually identify ${\ell }_{1^{*}}$ with $c_0$. We denote by ${\ell }_p^0$ and $c_0^0$ the linear span of ${\left\{e_n\right\}}_{n\in \omega }$ in ${\ell }_p$ or in $c_0$ endowed with the induced norm topology, respectively. We also use the following well-known description of relatively compact subsets of ${\ell }_p$ and $c_0$; see [33] (p. 6).

Proposition 1.

(i) A bounded subset A of ${\ell }_p$, $p\in [1,\infty )$, is relatively compact if and only if

$$\underset{m\to \infty }{lim}sup\left\{\sum _{m\le n}{\left|x_n\right|}^p:x=\left(x_n\right)\in A\right\}=0.$$

(ii) A bounded subset A of $c_0$ is relatively compact if and only if ${lim}_{n\to \infty }sup\left\{\right|x_n|:x=\left(x_n\right)\in A\}=0.$

We let $p\in [1,\infty ]$. A sequence ${\left\{x_n\right\}}_{n\in \omega }$ in a locally convex space E is called

• weakly p-convergent to $x\in E$ if ${\{x_n-x\}}_{n\in \omega }$ is weakly p-summable;
• weakly p-Cauchy if for each pair of strictly increasing sequences $\left(k_n\right),\left(j_n\right)\subseteq \omega$, the sequence ${(x_{k_n}-x_{j_n})}_{n\in \omega }$ is weakly p-summable.

Sequence ${\left\{{\chi }_n\right\}}_{n\in \omega }$ in $E^{\prime }$ is called weak* p-summable (resp., weak* p-convergent to $\chi \in E^{\prime }$ or weak* p-Cauchy) if it is weakly p-summable (resp., weakly p-convergent to $\chi \in E^{\prime }$ or weakly p-Cauchy) in $E_{w^{*}}^{\prime }$. Following [17], E is called p-barrelled (resp., p-quasibarrelled) if every weakly p-summable sequence in $E_{w^{*}}^{\prime }$ (resp., in $E_{\beta }^{\prime }$) is equicontinuous.

Generalizing the corresponding notions in the class of Banach spaces introduced in [18,24], the following p-versions of weakly compact-type properties are defined in [17]. We let $p\in [1,\infty ]$. A subset A of a locally convex space E is called

• (relatively) weakly sequentially p-compact if every sequence in A has a weakly p-convergent subsequence with limit in A (resp., in E);
• weakly sequentially p-precompact if every sequence from A has a weakly p-Cauchy subsequence.

An operator $T:E\to L$ between locally convex spaces E and L is called weakly sequentiallyp-(pre)compact if there is $U\in {\mathcal{N}}_0\left(E\right)$ such that $T\left(U\right)$ is a relatively weakly sequentially p-compact (resp., weakly sequentially p-precompact) subset of L.

The following class of subsets of an lcs E was introduced and studied in [17]. It generalizes the notion of p-$\left(V^{*}\right)$ subsets of Banach spaces defined in [34].

Definition 12.

We let $p,q\in [1,\infty ]$. A non-empty subset A of a locally convex space E is called a $(p,q)$-$\left(V^{*}\right)$ set (resp., a $(p,q)$-$\left(E{V}^{*}\right)$ set) if

$$\left(\underset{a\in A}{sup}\left|\langle {\chi }_n,a\rangle \right|\right)\in {\ell }_{q}ifq<\infty ,or\left(\underset{a\in A}{sup}\left|\langle {\chi }_n,a\rangle \right|\right)\in c_{0}ifq=\infty ,$$

for every (resp., equicontinuous) weakly p-summable sequence ${\left\{{\chi }_n\right\}}_{n\in \omega }$ in $E_{\beta }^{\prime }$. $(p,\infty )$-$\left(V^{*}\right)$ sets and $(1,\infty )$-$\left(V^{*}\right)$ sets are called simply p-$\left(V^{*}\right)$ sets and $\left(V^{*}\right)$ sets, respectively.

Let us recall $V^{*}$ type properties of Pełczyński defined in [17].

Definition 13.

We let $1\le p\le q\le \infty$. A locally convex space E is said to have

• the property $V_{(p,q)}^{*}$ (a property $E{V}_{(p,q)}^{*}$) if every $(p,q)$-$\left(V^{*}\right)$ (resp., $(p,q)$-$\left(E{V}^{*}\right)$) set in E is relatively weakly compact;
• the property $s{V}_{(p,q)}^{*}$ (property $sE{V}_{(p,q)}^{*}$) if every $(p,q)$-$\left(V^{*}\right)$ (resp., $(p,q)$-$\left(E{V}^{*}\right)$) set in E is relatively weakly sequentially compact;
• the weak property $s{V}_{(p,q)}^{*}$ or property $ws{V}_{(p,q)}^{*}$ (resp., weak property $sE{V}_{(p,q)}^{*}$ or property $wsE{V}_{(p,q)}^{*}$) if each $(p,q)$-$\left(V^{*}\right)$-subset (resp., $(p,q)$-$\left(E{V}^{*}\right)$-subset) of E is weakly sequentially precompact.

In the case when $q=\infty$, we omit subscript q and say that E has property $V_p^{*}$, etc., and in the case when $q=\infty$ and $p=1$, we say that E has property $V^{*}$, etc.

We let $p\in [1,\infty ]$. The p-Schur property of Banach spaces was defined in [35,36]. Generalizing this notion and following [17], an lcs E is said to have a p-Schur property if every weakly p-summable sequence is a null-sequence. In particular, E has the Schur property if and only if it is an ∞-Schur space. Following [32], E is called a weakly sequentially p-angelic space if the family of all relatively weakly sequentially p-compact sets in E coincides with the family of all relatively weakly compact subsets of E. The space E is a weakly p-angelic space if it is a weakly sequentially p-angelic space and each weakly compact subset of E is Fréchet–Urysohn.

Following [37], sequence $A={\left\{a_n\right\}}_{n\in \omega }$ in an lcs E is said to be equivalent to the standard unit basis $\{e_n:n\in \omega \}$ of ${\ell }_1$ if there exists a linear topological isomorphism R from $\overline{\mathsf{span}}\left(A\right)$ onto a subspace of ${\ell }_1$ such that $R\left(a_n\right)=e_n$ for every $n\in \omega$ (we do not assume that the closure $\overline{\mathsf{span}}\left(A\right)$ of the $\mathsf{span}\left(A\right)$ of A is complete or that R is onto). We also say that A is an ${\ell }_1$-sequence. Recall that a locally convex space E is said to have the Rosenthal property if every bounded sequence in E has a subsequence which either (1) is Cauchy in the weak topology or (2) is equivalent to the unit basis of ${\ell }_1$. The following remarkable extension of the celebrated Rosenthal ${\ell }_1$-theorem was proven by Ruess [29]: each locally complete locally convex space E whose every separable bounded set is metrizable has the Rosenthal property. Thus, every strict $\left(LF\right)$-space has the Rosenthal property.

Recall that an lcs X is called injective if for every subspace H of a locally convex space E, each operator $T:H\to X$ can be extended to operator $\overline{T}:E\to X$.

In [28], p. 377, Rosenthal pointed out a theorem of Odell and Stegall which states that any ∞-$\left(V^{*}\right)$ set of a Banach space is weakly sequentially precompact. In what follows, we use the following generalization of this remarkable result which is of independent interest (our detailed proof, follow the Odell–Stegall idea, cf. also Theorem 5.21 of [27]).

Theorem 4.

We let $2\le p\le q\le \infty$ and let E be a locally convex space with the Rosenthal property. Then, every $(p,q)$-$\left(V^{*}\right)$ subset of E is weakly sequentially precompact. Consequently, each $(p,q)$-limited subset of E is weakly sequentially precompact.

Proof. 

Suppose for a contradiction that there is a $(p,q)$-$\left(V^{*}\right)$ subset A of E which is not weakly sequentially precompact. So there is a sequence $S={\left\{x_n\right\}}_{n\in \omega }$ in A which does not have a weakly Cauchy subsequence. By the Rosenthal property of E and passing to a subsequence if needed, we can assume that S is an ${\ell }_1$-sequence. We set $H:=\overline{\mathsf{span}}\left(S\right)$ and let $P:H\to {\ell }_1$ be a topological isomorphism of H onto a subspace of ${\ell }_1$ such that $P\left(x_n\right)=e_n$ for every $n\in \omega$ (where ${\left\{e_n\right\}}_{n\in \omega }$ is the standard unit basis of ${\ell }_1$). We let $J:{\ell }_1\to {\ell }_p$, $I_1:{\ell }_1\to {\ell }_2$, and $I_2:{\ell }_2\to {\ell }_p$ be the natural inclusions; so $J=I_2\circ I_1$. By the Grothendieck Theorem 1.13 of [38], the operator $I_1$ is 1-summing. By the Ideal Property 2.4 of [38], J is also 1-summing, and hence, by the Inclusion Property 2.8 of [38], the operator J is 2-summing. By the discussion after Corollary 2.16 of [38], the operator J has the following factorization:

$$J:{\ell }_1\stackrel{R}{\to }{L}_{\infty }\left(\mu \right)\stackrel{J_2^{\infty }}{\to }{L}_2\left(\mu \right)\stackrel{Q}{\to }{\ell }_p,$$

where $\mu$ is a regular probability measure on some compact space K and $J_2^{\infty }:L_{\infty }\left(\mu \right)\to L_2\left(\mu \right)$ is the natural inclusion. By Theorem 4.14 of [38], the Banach space $L_{\infty }\left(\mu \right)$ is injective. Therefore, by Lemma 5.20 of [27], $L_{\infty }\left(\mu \right)$ is an injective locally convex space. In particular, operator $R\circ P:H\to L_{\infty }\left(\mu \right)$ can be extended to operator $T_{\infty }:E\to L_{\infty }\left(\mu \right)$. We set $T:=Q\circ J_2^{\infty }\circ T_{\infty }$. Then, T is an operator from E to ${\ell }_p$ such that

$$T\left(x_n\right)=Q\circ J_2^{\infty }\circ R\circ P\left(x_n\right)=J\circ P\left(x_n\right)=e_n\mathrm{for}\mathrm{every}n\in \omega ,$$

where ${\left\{e_n\right\}}_{n\in \omega }$ is the standard unit basis of ${\ell }_p$. Since A, and hence also S are $(p,q)$-$\left(V^{*}\right)$ sets, it follows that the canonical basis ${\left\{e_n\right\}}_{n\in \omega }$ of ${\ell }_p$ is also a $(p,q)$-$\left(V^{*}\right)$ set. However, this is impossible because the standard unit basis ${\left\{e_n^{*}\right\}}_{n\in \omega }$ of the dual ${\ell }_{p^{*}}$ is weakly p-summable (see Example 4.4 of [17]). Although, since $sup\left\{\right|\langle e_n^{*},e_i\rangle |:i\in \omega \}=1$ for all $n\in \omega$, it follows that ${\left\{e_n\right\}}_{n\in \omega }$ is not a $(p,q)$-$\left(V^{*}\right)$ set.

The last assertion follows from the easy fact that any $(p,q)$-limited set is a $(p,q)$-$\left(V^{*}\right)$ set. □

Motivated by the celebrated Rosenthal’s ${\ell }_1$ theorem, we introduce the following class of locally convex spaces.

Definition 14.

We let $p\in [1,\infty ]$. A locally convex space E is said to have the weak Cauchy subsequence property of order p (the $wCS{P}_p$ for short) if every bounded sequence in E has a weakly p-Cauchy subsequence. If $p=\infty$, we say simply that E has the $wCSP$.

Remark 1.

(i) By the Rosenthal ${\ell }_1$ theorem, a Banach space E has the $wCSP$ if and only if it has no isomorphic copy of ${\ell }_1$.

(ii) It is known (see Corollary 7.3.8 of [39]) that a Banach space E has no isomorphic copy of ${\ell }_1$ if and only if the dual Banach space $E_{\beta }^{\prime }$ has the weak Radon–Nikodym property, and hence if and only if E has the $wCSP$.

(iii) If $1\le p<q\le \infty$ and an lcs E has $wCS{P}_p$, then E has $wCS{P}_q$.

(iv) If an lcs E has $wCSP$, then E has the Rosenthal property.

(v) A Banach space E has $wCS{P}_p$ if and only if the identity map ${\mathsf{id}}_E:E\to E$ is weakly sequentially p-precompact.

Proposition 2.

If $1<p<\infty$, then ${\ell }_p$ has the $wCS{P}_p$ if and only if $p\ge 2$.

Proof. 

We assume that $1<p<2$. Then, $p<p^{*}$, and hence, by Proposition 6.5 of [17], ${\ell }_p$ has the p-Schur property. We let $S={\left\{e_n\right\}}_{n\in \omega }$ be the canonical unit basis of ${\ell }_p$. Assuming that S has a weakly p-Cauchy subsequence ${\left\{e_{n_k}\right\}}_{k\in \omega }$, the p-Schur property implies that it is Cauchy in ${\ell }_p$, which is impossible. Thus, ${\ell }_p$ does not have $wCS{P}_p$.

Assume that $2\le p<\infty$. Then, by Proposition 1.4 of [18] (or by Corollary 13.11 of [17]), $B_{{\ell }_p}$ is weakly sequentially $p^{*}$-compact. Since $1<p^{*}\le 2$, we have $p^{*}\le p$. Therefore, any weakly $p^{*}$-convergent sequence is also weakly p-convergent. Thus, $B_{{\ell }_p}$ is weakly sequentially p-compact, and hence ${\ell }_p$ has $wCS{P}_p$. □

Proposition 3.

We let $p\in [1,\infty ]$ and let ${\left\{E_i\right\}}_{i\in I}$ be a non-empty family of locally convex spaces.

(i)

If $I=\omega$, then $E={\prod }_{i\in \omega }{E}_i$ has the $wCS{P}_p$ if and only if each factor $E_i$ has the $wCS{P}_p$.

(ii)

The space $E={⨁}_{i\in I}{E}_i$ has the $wCS{P}_p$ if and only if each summand $E_i$ has the $wCS{P}_p$.

Proof. 

(i) We assume that E has the $wCS{P}_p$. Fix $j\in \omega$ and let $S={\left\{x_{n,j}\right\}}_{n\in \omega }$ be a bounded sequence in $E_j$. If ${\pi }_j$ is the natural embedding of $E_j$ into E, then the sequence ${\pi }_j\left(S\right)$ is bounded in E and hence, by $wCS{P}_p$, it has a weakly p-Cauchy subsequence ${\left\{{\pi }_j\left(x_{n_k,j}\right)\right\}}_{k\in \omega }$. It is clear that ${\left\{x_{n_k,j}\right\}}_{k\in \omega }$ is weakly p-Cauchy in $E_j$. Thus, $E_j$ has $wCS{P}_p$.

Conversely, we assume that all spaces $E_i$ have $wCS{P}_p$ and let ${\{x_n={\left(x_{n,i}\right)}_{i\in \omega }\}}_{n\in \omega }$ be a sequence in E. We proceed by induction on $i\in \omega$. For $i=0$, $wCS{P}_p$ of $E_0$ implies that there is sequence $I_0$ in $\omega$ such that sequence ${\left\{x_{j,0}\right\}}_{j\in I_0}$ is weakly p-Cauchy in $E_0$. For $i=1$, the $wCS{P}_p$ of $E_1$ implies that there is subsequence $I_1$ of $I_0$ such that sequence ${\left\{x_{j,1}\right\}}_{j\in I_1}$ is weakly p-Cauchy in $E_1$. Continuing this process, we find sequence $\omega \supseteq I_0\supseteq I_1\supseteq \cdots$ such that sequence ${\left\{x_{j,i}\right\}}_{j\in I_i}$ is weakly p-Cauchy in $E_i$ for every $i\in \omega$. For every $i\in \omega$, we choose $m_i\in I_i$ such that $m_i<m_{i+1}$ for all $i\in \omega$. We claim that subsequence ${\left\{x_{m_i}\right\}}_{i\in \omega }$ of ${\left\{x_n\right\}}_{n\in \omega }$ is weakly p-Cauchy in E. Indeed, since $E^{\prime }={⨁}_{i\in \omega }{E}_i^{\prime }$, each $\chi \in E^{\prime }$ has a form $\chi =({\chi }_0,\cdots ,{\chi }_k,0,\dots )$. Then, for every strictly increasing sequence $\left(i_n\right)\subseteq \omega$, we have

$$|\langle \chi ,x_{m_{i_n}}-x_{m_{i_{n+1}}}\rangle |\le \sum _{t=0}^k|\langle {\chi }_t,x_{m_{i_n},t}-x_{m_{i_{n+1}},t}\rangle |$$

(1)

By the choice of $I_t$ and $\left(m_i\right)\in \omega$, each sequence ${\left\{x_{m_{i_n},t}-x_{m_{i_{n+1}},t}\right\}}_{n\in \omega }$ is weakly p-summable. This and (1) imply that sequence ${\left\{x_{m_{i_n}}-x_{m_{i_{n+1}}}\right\}}_{n\in \omega }$ is weakly p-summable, and hence ${\left\{x_{m_i}\right\}}_{i\in \omega }$ is weakly p-Cauchy in E. Thus, E has $wCS{P}_p$.

(ii) The necessity can be proven repeating word for word the necessity in (i). The sufficiency follows from (i) because any bounded sequence in E has finite support. □

Remark 2.

The countability of the index set in (i) of Proposition 3 is essential. Indeed, by the example in Lemma 2.2 of [17], product ${\mathbb{R}}^{\mathfrak{c}}$ has a bounded sequence without weak Cauchy subsequences.

3. Permanent Properties of Gelfand–Phillips Type Properties

In this section, we study relationships between different Gelfand–Phillips type properties, stability under taking direct products and direct sums, and a connection of these properties with the Schur property. We also show that the class of locally convex spaces with the Gelfand–Phillips property is sufficiently large.

Recall that a locally convex space E is von Neumann complete if every precompact subset of E is relatively compact. Following general Definition 11.12 of [17], E is ${\mathsf{L}}_{(p,q)}$-von Neumann complete (resp., ${\mathsf{CL}}_{(p,q)}$-von Neumann complete) if every closed precompact set in ${\mathsf{L}}_{(p,q)}$ (resp., in ${\mathsf{CL}}_{(p,q)}$) is compact. Recall also that a locally convex space E is called semi-Montel if every bounded subset of E is relatively compact, and E is a Montel space if it is a barrelled semi-Montel space.

Lemma 2.

We let $1\le p\le q\le \infty$ and let $(E,\tau )$ be a locally convex space.

(i)

If E has the $EG{P}_{(p,q)}$ (resp., $sEG{P}_{(p,q)}$) property, then E has the $G{P}_{(p,q)}$ (resp., $sG{P}_{(p,q)}$) property. The converse is true for p-barrelled spaces.

(ii)

If $1\le p^{\prime }\le p$ and $q\le q^{\prime }\le \infty$ and E has the $G{P}_{(p^{\prime },q^{\prime })}$ (resp., $sG{P}_{(p^{\prime },q^{\prime })}$, $EG{P}_{(p^{\prime },q^{\prime })}$, $sEG{P}_{(p^{\prime },q^{\prime })}$, $prG{P}_{(p^{\prime },q^{\prime })}$, or $spG{P}_{(p^{\prime },q^{\prime })}$) property, then E has the $G{P}_{(p,q)}$ (resp., $sG{P}_{(p,q)}$, $EG{P}_{(p,q)}$, $sEG{P}_{(p,q)}$, $prG{P}_{(p,q)}$, or $spG{P}_{(p,q)}$) property.

(iii)

If the class of relatively compact sets in E coincides with class of relatively sequentially compact sets, then E has property $G{P}_{(p,q)}$ (resp., $EG{P}_{(p,q)}$) if and only if it has property $sG{P}_{(p,q)}$ (resp., $sEG{P}_{(p,q)}$).

(iv)

If E is an angelic p-barrelled space, then all the properties $G{P}_{(p,q)}$, $EG{P}_{(p,q)}$, $sG{P}_{(p,q)}$ and $sEG{P}_{(p,q)}$ coincide for E.

(v)

If E has the $G{P}_{(p,q)}$ property, then it has the $prG{P}_{(p,q)}$ property. The converse is true if E is ${\mathsf{L}}_{(p,q)}$-von Neumann complete.

(vi)

If E is sequentially complete, then E has the $spG{P}_{(p,q)}$ (resp., $spEG{P}_{(p,q)}$) property if and only if it has the $sG{P}_{(p,q)}$ (resp., $sEG{P}_{(p,q)}$) property.

(vii)

If E is semi-Montel, then E has the $EG{P}_{(p,q)}$ property.

Proof. 

(i) follows from the easy fact that every $(p,q)$-limited set is $(p,q)$-$\mathcal{E}$-limited; see Lemma 3.1(i) of [27] and the definition of p-barrelled spaces.

(ii) follows from the fact that every $(p,q)$-($\mathcal{E}$-)limited set is $(p^{\prime },q^{\prime })$-($\mathcal{E}$-)limited; see Lemma 3.1(vi) of [27].

(iii), (v), and (vii) follow from the corresponding definitions.

(iv) follows from (i) and the fact that for angelic spaces, relatively compact sets are exactly relatively sequentially compact sets.

(vi) suffices to prove the necessity. We assume that E has the $spG{P}_{(p,q)}$ (resp., $spEG{P}_{(p,q)}$) property. We let A be a $(p,q)$-limited (resp., $(p,q)$-$\mathcal{E}$-limited) set in E. Then, A is sequentially precompact. We show that A is even relatively sequentially compact. Indeed, we let $S={\left\{x_n\right\}}_{n\in \omega }$ be a sequence in A. Since A is sequentially precompact, S has a Cauchy subsequence ${\left\{x_{n_k}\right\}}_{k\in \omega }$. Since E is sequentially complete, there is $x\in E$ such that $x_{n_k}\to x$. Therefore, A is relatively sequentially compact. Thus, E has the $sG{P}_{(p,q)}$ (resp., $sEG{P}_{(p,q)}$) property. □

If space E carries its weak topology (for example, $E=C_p\left(X\right)$), we have the following result:

Lemma 3.

We let $1\le p\le q\le \infty$ and let $(E,\tau )$ be a locally convex space such that $E=E_w$. Then,

(i)

E has the $prEG{P}_{(p,q)}$ property and hence the $prG{P}_{(p,q)}$ property;

(ii)

every bounded subset of E is $(p,q)$-$\mathcal{E}$-limited; consequently, if E is von Neumann complete, then E has property $EG{P}_{(p,q)}$.

Proof. 

(i) is trivial because every bounded subset of E is precompact.

(ii) We let A be a bounded subset of E and let $S={\left\{{\chi }_n\right\}}_{n\in \omega }$ be an equicontinuous weak* p-summable sequence in $E^{\prime }$. Since E carries its weak topology, it follows that $S\subseteq F^{\circ \circ }$ for some finite subset $F=\{{\eta }_1,\dots ,{\eta }_k\}\subseteq E^{\prime }$. Therefore, $S\subseteq \mathsf{span}\left(F\right)$, and hence for every $n\in \omega$, there are scalars $c_{1,n},\dots ,c_{k,n}\in \mathbb{F}$ such that

$${\chi }_n=c_{1,n}{\eta }_1+\cdots +c_{k,n}{\eta }_k.$$

Since S is weak* p-summable, we determine that ${\left(c_{i,n}\right)}_n\in {\ell }_p$ (or $\in c_0$ if $p=\infty$) for every $i=1,\dots ,k$. As A is bounded, for every $i=1,\dots ,k$, we can define $C_i:={sup}_{a\in A}|\langle {\eta }_i,a\rangle |$. Then,

$$\underset{a\in A}{sup}|\langle {\chi }_n,a\rangle |\le \sum _{i=1}^k\underset{a\in A}{sup}|c_{i,n}\left|\right|\langle {\eta }_i,a\rangle |=\sum _{i=1}^k{C}_i\left|c_{i,n}\right|.$$

Therefore, ${\left({\sup }_{a\in A}\left|\langle {\chi }_n,a\rangle \right|\right)}_n\in {\ell }_p\subseteq {\ell }_q$ (or $\in c_0$ if $p=q=\infty$). Thus, A is a $(p,q)$-$\mathcal{E}$-limited set. □

Note that by Theorem 3.13 of [17], $C_p\left(X\right)$ is von Neumann complete if and only if X is discrete.

Lemma 4.

We let $1\le p\le q\le \infty$ and let τ be a locally convex compatible topology on a locally convex space $(E,\mathcal{T})$ such that $\tau \subseteq \mathcal{T}$. If $(E,\mathcal{T})$ has the $prG{P}_{(p,q)}$ (resp., $spG{P}_{(p,q)}$, $G{P}_{(p,q)}$, or $sG{P}_{(p,q)}$) property; then, $(E,\tau )$ also has the same property.

Proof. 

Since $\tau$ and $\mathcal{T}$ are compatible, spaces $(E,\tau )$ and $(E,\mathcal{T})$ have the same $(p,q)$-limited subsets by Lemma 3.1(vii) of [27]. As by assumption, all $(p,q)$-limited sets in $(E,\mathcal{T})$ are precompact (resp., sequentially precompact, relatively compact or relatively sequentially compact), inclusion $\tau \subseteq \mathcal{T}$ implies that so are all $(p,q)$-limited sets in $(E,\tau )$. □

Remark 3.

The converse in Lemma 4 is not true in general. Indeed, we let $E={\ell }_{\infty }$. Then, $E_w$ trivially has the precompact $GP$ property. However, space E is not a Gelfand–Phillips space by [2]. Consequently, the property of being a Gelfand–Phillips space is not a property of dual pair $(E,E^{\prime })$.

Below, we consider relationships between the b-version of the Gelfand–Phillips properties and the Gelfand–Phillips properties.

Lemma 5.

We let $1\le p\le q\le \infty$ and let $(E,\tau )$ be a locally convex space. Then,

(i)

if E has the b-$G{P}_{(p,q)}$ property, then it has the $prG{P}_{(p,q)}$ property;

(ii)

if E is barrelled, then E has the b-$G{P}_{(p,q)}$ (resp., b-$EG{P}_{(p,q)}$) property if and only if it has the $prG{P}_{(p,q)}$ (resp., $prEG{P}_{(p,q)}$) property;

(iii)

if E is barrelled and von Neumann complete, then E has the b-$G{P}_{(p,q)}$ (resp., b-$EG{P}_{(p,q)}$) property if and only if it has the $G{P}_{(p,q)}$ (resp., $EG{P}_{(p,q)}$) property.

Proof. 

(i) follows from inclusion $\tau \subseteq \beta (E,E^{\prime })$.

(ii) and (iii) follow from equality $E=E_{\beta }$ and the corresponding definitions. □

We select the next proposition which shows that for wide classes of locally convex spaces important for applications, in fact, there is only a unique Gelfand–Phillips type property (however, in general, these notions are different; see Examples 1 and 2 and Theorem 5 below).

Proposition 4.

We let $1\le p\le q\le \infty$ and let E be an angelic, complete and barrelled space (for example, E is a strict $\left(LF\right)$-space). Then, all the properties $G{P}_{(p,q)}$, $EG{P}_{(p,q)}$, $sG{P}_{(p,q)}$, $sEG{P}_{(p,q)}$, $prG{P}_{(p,q)}$, $prEG{P}_{(p,q)}$, $spG{P}_{(p,q)}$, $spEG{P}_{(p,q)}$ and b-$G{P}_{(p,q)}$ are equivalent.

Proof. 

The properties $G{P}_{(p,q)}$, $EG{P}_{(p,q)}$, $sG{P}_{(p,q)}$, and $sEG{P}_{(p,q)}$ are equivalent by (iv) of Lemma 2. The properties $G{P}_{(p,q)}$ and $prG{P}_{(p,q)}$ are equivalent by (v) of Lemma 2. The properties $sG{P}_{(p,q)}$ and $spG{P}_{(p,q)}$ are equivalent by (vi) of Lemma 2. The properties $prG{P}_{(p,q)}$ and b-$G{P}_{(p,q)}$ are equivalent by (ii) of Lemma 5.

Since E is barrelled, the $(p,q)$-limited sets and the $(p,q)$-$\mathcal{E}$-limited sets are the same. Therefore, properties $prG{P}_{(p,q)}$ and $prEG{P}_{(p,q)}$, as well as properties $spG{P}_{(p,q)}$ and $spEG{P}_{(p,q)}$, coincide. □

By Proposition 4.2 of [27], every p-limited subset of E is coarse p-limited. Although every precompact subset of a $c_0$-barrelled space is limited by Corollary 3.7 of [27], non-$c_0$-barrelled spaces may contain even convergent sequences which are not limited; see Example 3.10 in [27]. We complement these results in the the following assertion. Recall that a subset A of a topological space X is functionally bounded if $f\left(A\right)$ is bounded for every $f\in C\left(X\right)$; clearly, any compact subset of X is functionally bounded.

Proposition 5.

We let $p\in [1,\infty ]$ and let E be a locally convex space. Then,

(i)

each functionally bounded subset A of E is coarse p-limited;

(ii)

every limited subset of E is coarse p-limited.

Proof. 

(i) We let $T:E\to {\ell }_p$ (or $T:E\to c_0$ if $p=\infty$) be an operator. Then, $T\left(A\right)$ is a functionally bounded subset of the metric space ${\ell }_p$ (or $c_0$), and hence $T\left(A\right)$ is relatively compact. Thus, A is coarse p-limited.

(ii) We let A be a limited subset of E and let $T\in \mathcal{L}(E,{\ell }_p)$ (or $T\in \mathcal{L}(E,c_0)$ if $p=\infty$). Then, by (iv) of Lemma 3.1 of [27], $T\left(A\right)$ is a limited subset of ${\ell }_p$ (or $c_0$). Since the Banach spaces ${\ell }_p$ and $c_0$ are separable, they have the Gelfand–Phillips property by Theorem 2. Therefore, $T\left(A\right)$ is relatively compact in ${\ell }_p$ (or in $c_0$). Thus, A is a coarse p-limited set in E. □

Below, we consider the coarse versions of the Gelfand–Phillips property.

Proposition 6.

We let $1\le p\le q\le \infty$ and let $(E,\tau )$ be a locally convex space.

(i)

If E has the coarse $G{P}_p$ (resp., coarse $sG{P}_p$ or coarse $spG{P}_p$) property, then E has the coarse $prG{P}_p$ (resp., coarse $spG{P}_p$ or coarse $prG{P}_p$) property.

(ii)

If $E=E_w$, then ${\mathsf{CL}}_p\left(E\right)=\mathsf{Bo}\left(E\right)$ and hence E has the coarse $prG{P}_p$ property.

(iii)

If E has the coarse $G{P}_p$ (resp., coarse $sG{P}_p$, coarse $prG{P}_p$, or coarse $spG{P}_p$) property, then E has the $G{P}_p$ (resp., $sG{P}_p$, $prG{P}_p$, or $spG{P}_p$) property.

(iv)

If E has the coarse $G{P}_p$ property, then E has the $GP$ property. The converse is not true in general.

(v)

If E is barrelled, then E has the $G{P}_{(1,\infty )}$ (resp., $sG{P}_{(1,\infty )}$, $prG{P}_{(1,\infty )}$, or $spG{P}_{(1,\infty )}$) property if and only if it has the coarse $G{P}_1$ (resp., coarse $sG{P}_1$, coarse $prG{P}_1$, or coarse $spG{P}_1$) property.

Proof. 

(i) is obvious.

(ii) Since $E=E_w$, Lemma 5.12 of [27] implies that each operator from E into any Banach space is finite-dimensional. Therefore, any bounded subset of E is coarse p-limited. As any bounded subset of E is precomapct, it follows that E has the coarse $prG{P}_p$ property.

(iii) follows from Proposition 4.2 of [27] which states that every p-limited set is coarse p-limited.

(iv) follows from Proposition 5. To show that the converse is not true in general, we let $E=c_0$ and $1\le p<\infty$. Since E is separable, it has the $GP$ property by Theorem 2. However, E does not have the coarse $G{P}_p$ property because, by the Pitt theorem, $B_E$ is a non-compact coarse p-limited set.

(v) follows from Proposition 3.13 of [27] which states that for a barrelled space E, the class of $(1,\infty )$-limited sets coincides with the class of coarse 1-limited sets. □

It is natural to consider relationships between $V_{(p,q)}^{*}$ type properties of Pełczyński and Gelfand–Phillips type properties. Following [32], an lcs E has the (weak) Glicksberg property if E and $E_w$ have the same (resp., absolutely convex) compact sets. Note that the weak Glicksberg property does not imply the Schur property and vise versa.

Proposition 7.

We let $1\le p\le q\le \infty$ and let $(E,\tau )$ be a locally convex space.

(i)

We let E have the weak Glicksberg property. If E has the property $V_{(p,q)}^{*}$ (resp., $E{V}_{(p,q)}^{*}$), then E has the $G{P}_{(p,q)}$ (resp., $EG{P}_{(p,q)}$) property.

(ii)

We let E have the Schur property. If E has the property $s{V}_{(p,q)}^{*}$ (resp., $sE{V}_{(p,q)}^{*}$), then E has the $sG{P}_{(p,q)}$ (resp., $sEG{P}_{(p,q)}$) property.

Proof. 

(i) Recall that every $(p,q)$-limited (resp., $(p,q)$-$\mathcal{E}$-limited) set A of E is a also a $(p,q)$-$\left(V^{*}\right)$ (resp., $(p,q)$-$\left(E{V}^{*}\right)$) set by Lemma 3.1(viii) of [27]. By (iii) of Lemma 7.2 of [17], $A^{\circ \circ }$ is also a $(p,q)$-$\left(V^{*}\right)$ (resp., $(p,q)$-$\left(E{V}^{*}\right)$) set. By property $V_{(p,q)}^{*}$ (resp., $E{V}_{(p,q)}^{*}$), $A^{\circ \circ }$ is weakly compact, and hence, by the weak Glicksberg property, $A^{\circ \circ }$ is a compact subset of E. Therefore, A is relatively compact. Thus, E has the $G{P}_{(p,q)}$ (resp., $EG{P}_{(p,q)}$) property.

(ii) We let A be a $(p,q)$-limited (resp., $(p,q)$-$\mathcal{E}$-limited) set in E. Then, by Lemma 3.1(viii) of [27], A is a $(p,q)$-$\left(V^{*}\right)$ (resp., $(p,q)$-$\left(E{V}^{*}\right)$) set. By property $s{V}_{(p,q)}^{*}$ (resp., $sE{V}_{(p,q)}^{*}$), A is relatively weakly sequentially compact. Since E is a Schur space, Lemma 2.1 of [27] implies that A is relatively sequentially compact in E. Thus, E has the $sG{P}_{(p,q)}$ (resp., $sEG{P}_{(p,q)}$) property. □

Remark 4.

In both cases (i) and (ii) of Proposition 7, the converse is not true in general. Indeed, we let $E=c_0$ and $p=q=\infty$. Being separable, $c_0$ has the $GP$ property and hence, by Proposition 4, the $sGP$ property. On the other hand, E has neither property $V_{\infty }^{*}$ nor property $s{V}_{\infty }^{*}$ because the Schur property of $E_{\beta }^{\prime }={\ell }_1$ implies that $B_E$ is a non-compact ∞-$\left(V^{*}\right)$ set.

Below, we show that the classes of locally convex spaces with Gelfand–Phillips type properties are stable under taking direct products and direct sums.

Proposition 8.

We let $1\le p\le q\le \infty$ and let ${\left\{E_i\right\}}_{i\in I}$ be a non-empty family of locally convex spaces. Then,

(i)

$E={\prod }_{i\in I}{E}_i$ has the $G{P}_{(p,q)}$ (resp., $EG{P}_{(p,q)}$, $prG{P}_{(p,q)}$, $prEG{P}_{(p,q)}$, coarse $G{P}_p$, or coarse $prG{P}_p$) property if and only if all spaces $E_i$ have the same property;

(ii)

$E={⨁}_{i\in I}{E}_i$ has the $G{P}_{(p,q)}$ (resp., $EG{P}_{(p,q)}$, $sG{P}_{(p,q)}$, $sEG{P}_{(p,q)}$, $prG{P}_{(p,q)}$, $prEG{P}_{(p,q)}$, $spG{P}_{(p,q)}$, $spEG{P}_{(p,q)}$, coarse $G{P}_p$, coarse $sG{P}_p$, coarse $spG{P}_p$, or coarse $prG{P}_p$) property if and only if all spaces $E_i$ have the same property;

(iii)

if $I=\omega$ is countable, then $E={\prod }_{i\in \omega }{E}_i$ has the $sG{P}_{(p,q)}$ (resp., $sEG{P}_{(p,q)}$, $spG{P}_{(p,q)}$, $spEG{P}_{(p,q)}$, coarse $sG{P}_p$, or coarse $spG{P}_p$) property if and only if all spaces $E_i$ have the same property.

Proof. 

We consider only $G{P}_{(p,q)}$ type properties because the coarse $G{P}_p$ type properties can be considered analogously using properties of coarse p-limited sets; see Lemma 4.1 and Proposition 4.3 of [27].

To prove the necessity, we let E have the $G{P}_{(p,q)}$ (resp., $EG{P}_{(p,q)}$, $sG{P}_{(p,q)}$, $sEG{P}_{(p,q)}$, $prG{P}_{(p,q)}$, $prEG{P}_{(p,q)}$, $spG{P}_{(p,q)}$ or $spEG{P}_{(p,q)}$) property. We fix $j\in I$ and let $A_j$ be a $(p,q)$-limited (resp., $(p,q)$-$\mathcal{E}$-limited) set in $E_j$. Since $E_j$ is a direct summand of E, to show that $A_j$ is relatively compact (resp., precompact, relatively sequentially compact or sequentially precompact) in $E_j$, it suffices to show that the set $A:=A_j\times {\prod }_{i\in I\setminus \left\{j\right\}}\left\{0_i\right\}$ has the same property in E. In all cases (i)–(iii), by Proposition 3.3 of [27], A is a $(p,q)$-limited (resp., $(p,q)$-$\mathcal{E}$-limited) set in E (also as the image of $A_j$ under the canonical embedding of $E_j$ into E). Since E has the $G{P}_{(p,q)}$ (resp., $EG{P}_{(p,q)}$, $sG{P}_{(p,q)}$, $sEG{P}_{(p,q)}$, $prG{P}_{(p,q)}$, $prEG{P}_{(p,q)}$, $spG{P}_{(p,q)}$ or $spEG{P}_{(p,q)}$) property, it follows that A is relatively compact (resp., precompact, relatively sequentially compact or sequentially precompact) in E, as desired.

To prove sufficiency, we assume that all spaces $E_i$ have the $G{P}_{(p,q)}$ (resp., $EG{P}_{(p,q)}$, $sG{P}_{(p,q)}$, $sEG{P}_{(p,q)}$, $prG{P}_{(p,q)}$, $prEG{P}_{(p,q)}$, $spG{P}_{(p,q)}$ or $spEG{P}_{(p,q)}$) property, and we let A be a $(p,q)$-limited (resp., $(p,q)$-$\mathcal{E}$-limited) set in E. Recall that Lemma 2.1 of [17] states that a subset of a countable product of Tychonoff spaces is (relatively) sequentially compact if and only if so are all its projections. This result and, for Cases (i) and (iii), the fact that the product of relatively compact sets (resp., precompact sets) is relatively compact (resp., precompact), and for Case (ii) the fact that the support of A is finite since A is bounded to show that A is relatively compact (resp., precompact, relatively sequentially compact or sequentially precompact), it suffices to prove that for every $i\in I$, the projection $A_i$ of A onto the ith coordinate has the same property in $E_i$. But this condition is satisfied because, by (iv) of Lemma 3.1 of [27], projection $A_i$ is a $(p,q)$-limited (resp., $(p,q)$-$\mathcal{E}$-limited) set in $E_i$ and hence it is relatively compact (resp., precompact, relatively sequentially compact or sequentially precompact) in $E_i$ by the corresponding property of $E_i$. □

In (iii) of Proposition 8, the countability of set I is essential.

Example 1.

We let $1\le p\le q\le \infty$. Then, the separable space ${\mathbb{R}}^{\mathfrak{c}}$ has the $EG{P}_{(p,q)}$ property and hence the $G{P}_{(p,q)}$ property and the coarse $prG{P}_p$ property, but it has neither the $spG{P}_{(p,q)}$ property nor the coarse $spG{P}_p$ property.

Proof. 

Space ${\mathbb{R}}^{\mathfrak{c}}$ has the $EG{P}_{(p,q)}$ property and the coarse $prG{P}_p$ property by (i) of Proposition 8. By (i) of Proposition 3.3 of [27], any bounded subset A of ${\mathbb{R}}^{\mathfrak{c}}$ is $(p,q)$-limited (and hence also $(p,q)$-$\mathcal{E}$-limited) and, by (i) of Proposition 4.3 of [27], A is also coarse p-limited. In particular, identifying $3^{\omega }$ with $\mathfrak{c}$ we determine that sequence $S={\left\{f_n\right\}}_{n\in \omega }$ of functions constructed in Lemma 2.2 of [17] is a $(p,q)$-limited set. However, it is proven in Lemma 2.2 of [17] that S is not sequentially precompact. Thus, ${\mathbb{R}}^{\mathfrak{c}}$ has neither the $spG{P}_{(p,q)}$ property nor the coarse $spG{P}_p$ property. □

In the next proposition, we consider the heredity of Gelfand–Phillips type properties. Recall that subspace Y of a Tychonoff space X is sequentially closed if Y contains all limits of convergent sequences from Y.

Proposition 9.

We let $1\le p\le q\le \infty$ and let L be a subspace of a locally convex space E.

(i)

If L is closed in E and E has the $G{P}_{(p,q)}$ (resp., $EG{P}_{(p,q)}$ or coarse $G{P}_p$) property, then L also has the same property.

(ii)

If L is sequentially closed in E and E has the $sG{P}_{(p,q)}$ (resp., $sEG{P}_{(p,q)}$ or coarse $sG{P}_p$) property, then L also has the same property.

(iii)

If E has the $prG{P}_{(p,q)}$ (resp., $prEG{P}_{(p,q)}$ or coarse $prG{P}_p$) property, then L also has the same property.

(iv)

If E has the $spG{P}_{(p,q)}$ (resp., $spEG{P}_{(p,q)}$ or coarse $spG{P}_p$) property, then L also has the same property.

Proof. 

Below, we consider only $G{P}_{(p,q)}$ type properties because the coarse $G{P}_p$ type properties can be considered analogously using properties of coarse p-limited sets; see Lemma 4.1 of [27].

We let A be a $(p,q)$-limited (resp., $(p,q)$-$\mathcal{E}$-limited) set in L. Then, by (iv) of Lemma 3.1 of [27] applied to the identity embedding ${\mathsf{id}}_L:L\to E$, A is a $(p,q)$-limited (resp., $(p,q)$-$\mathcal{E}$-limited) also set in E.

(i) By property $G{P}_{(p,q)}$ (resp., $EG{P}_{(p,q)}$) of E, set A is relatively compact in E. Since L is a closed subspace of E, it follows that A is relatively compact in L. Thus, L has property $G{P}_{(p,q)}$ (resp., $EG{P}_{(p,q)}$).

(ii) By property $sG{P}_{(p,q)}$ (resp., $sEG{P}_{(p,q)}$) of E, set A is relatively sequentially compact in E. Since L is a sequentially closed subspace of E, it follows that A is relatively sequentially compact in L. Thus, L has property $sG{P}_{(p,q)}$ (resp., $sEG{P}_{(p,q)}$).

(iii) By property $prG{P}_{(p,q)}$ (resp., $prEG{P}_{(p,q)}$), set A is precompact in E and hence also in L. Thus, L has property $prG{P}_{(p,q)}$ (resp., $prEG{P}_{(p,q)}$).

(iv) By property $spG{P}_{(p,q)}$ (resp., $spEG{P}_{(p,q)}$) of E, set A is sequentially precompact in E. We let ${\left\{a_n\right\}}_{n\in \omega }$ be a sequence in A. We take subsequence ${\left\{a_{n_k}\right\}}_{k\in \omega }$ of ${\left\{a_n\right\}}_{n\in \omega }$ which is Cauchy in E. Since L is a subspace of E, it follows that ${\left\{a_{n_k}\right\}}_{k\in \omega }$ is also Cauchy in L. Therefore, A is sequentially precompact in L. Thus, L has property $spG{P}_{(p,q)}$ (resp., $spEG{P}_{(p,q)}$). □

Below, we characterize spaces $C_p\left(X\right)$ which have one of the equicontinuous $GP$ properties.

Theorem 5.

We let $1\le p\le q\le \infty$, and let X be a Tychonoff space. Then,

(i)

$C_p\left(X\right)$ has the $EG{P}_{(p,q)}$ property if and only if X is discrete;

(ii)

$C_p\left(X\right)$ has the b-$EG{P}_{(p,q)}$ property if and only if every functionally bounded subset of X is finite;

(iii)

$C_p\left(X\right)$ has the $prEG{P}_{(p,q)}$ property and hence the $prG{P}_{(p,q)}$ property.

Proof. 

(i) We assume that $C_p\left(X\right)$ has the $EG{P}_{(p,q)}$ property. Then, by (ii) of Lemma 3, the bounded set $B=\{f\in C_p\left(X\right):\parallel f{\parallel }_X\le 1\}$ is $(p,q)$-$\mathcal{E}$-limited. Therefore, by the $EG{P}_{(p,q)}$ property, B has compact closure in $C_p\left(X\right)$. As B is dense in compact space ${\mathbb{D}}^X$, we determine that ${\mathbb{D}}^X\subseteq C_p\left(X\right)$. But this is possible only if X is discrete.

Conversely, we assume that X is discrete. Then, clearly, every bounded subset of $C_p\left(X\right)=\mathbb{F}X$ has compact closure. Thus, $C_p\left(X\right)$ has the $EG{P}_{(p,q)}$ property.

(ii) We assume that $C_p\left(X\right)$ has the b-$EG{P}_{(p,q)}$ property and let A be a functionally bounded subset of X. Assuming that A is infinite, we can find sequence $\left\{a_n\right\}\subseteq A$ and sequence ${\left\{U_n\right\}}_{n\in \omega }$ of open subsets of X such that $a_n\in U_n$ and $U_n\cap U_m=\varnothing$ for all distinct $n,m\in \omega$. For every $n\in \omega$, we choose $f_n\in C_p\left(X\right)$ such that $f_n(X\setminus U_n)=\left\{0\right\}$ and $f_n\left(a_n\right)=n$. Then, sequence $S={\left\{f_n\right\}}_{n\in \omega }$ is a bounded subset of $C_p\left(X\right)$ and hence, by (ii) of Lemma 3, S is a $(p,q)$-$\mathcal{E}$-limited set. Therefore, to obtain a contradiction with the b-$EG{P}_{(p,q)}$ property of $C_p\left(X\right)$, it suffices to show that S is not barrel-precompact. To this end, we note that the functional boundedness of A implies that set $B:=\{f\in C_p\left(X\right):\parallel f{\parallel }_A\le 1\}$ is a barrel in $C_p\left(X\right)$. Since by construction, $S⊈kB$ for every $k\in \omega$, we determine that S is not barrel-precompact.

Conversely, we assume that every functionally bounded subset of X is finite. We show that every $(p,q)$-$\mathcal{E}$-limited subset B of $C_p\left(X\right)$ is barrel-precompact. To this end, we note that, by the Buchwalter–Schmetz theorem, space $C_p\left(X\right)$ is barrelled. Now, we let D be a barrel in $C_p\left(X\right)$. Then, D is a neighborhood of zero. Since $C_p\left(X\right)$ carries its weak topology and B is its bounded subset, it follows that B is precompact and hence, there is a finite subset $F\subseteq C_p\left(X\right)$ such that $B\subseteq F+D$. Thus, B is barrel-precompact.

(iii) Since $C_p\left(X\right)$ carries its weak topology, the assertion follows from (i) of Lemma 3. □

In (i) and (ii) of Proposition 9, the condition on L of being a (sequentially) closed subspace of E is essential even for Fréchet spaces as the next example shows.

Example 2.

We let $1\le p\le q\le \infty$ and let X be a countable non-discrete Tychonoff space whose compact subsets are finite. Then, the dense subspace $C_p\left(X\right)$ of ${\mathbb{F}}^{\omega }$ has none of the $G{P}_{(p,q)}$, $EG{P}_{(p,q)}$, $sG{P}_{(p,q)}$, or $sEG{P}_{(p,q)}$ properties, but it has the $prEG{P}_{(p,q)}$, $prG{P}_{(p,q)}$, $spEG{P}_{(p,q)}$ and $spG{P}_{(p,q)}$ properties.

Proof. 

By the Buchwalter–Schmetz theorem, space $C_p\left(X\right)$ is barrelled and hence it is p-barrelled. Being metrizable, $C_p\left(X\right)$ is an angelic space. Thus, by (iv) of Lemma 2 and (i) of Theorem 5, $C_p\left(X\right)$ has none of properties $G{P}_{(p,q)}$, $EG{P}_{(p,q)}$, $sG{P}_{(p,q)}$, or $sEG{P}_{(p,q)}$.

On the other hand, by (iii) of Theorem 5 and the metrizability of ${\mathbb{F}}^{\omega }$, space $C_p\left(X\right)$ has properties $prEG{P}_{(p,q)}$, $prG{P}_{(p,q)}$, $spEG{P}_{(p,q)}$ and $spG{P}_{(p,q)}$. □

Our next aim is to show that any locally convex space is a quotient space of a locally convex space with the (sequential) $G{P}_{(p,q)}$ property for all $1\le p\le q\le \infty$. First, we recall some definitions.

Following [40], the free locally convex space $L\left(X\right)$ over a Tychonoff space X is a pair consisting of a locally convex space $L\left(X\right)$ and a continuous map $i:X\to L\left(X\right)$ such that every continuous map f from X to a locally convex space E gives rise to a unique continuous linear operator ${\Psi }_E\left(f\right):L\left(X\right)\to E$ with $f={\Psi }_E\left(f\right)\circ i$. The free locally convex space $L\left(X\right)$ always exists and is essentially unique. For $\chi =a_1{x}_1+\cdots +a_n{x}_n\in L\left(X\right)$ with distinct $x_1,\dots ,x_n\in X$ and nonzero $a_1,\dots ,a_n\in \mathbb{F}$, we set $\chi \left[x_i\right]:=a_i$ and

$$\parallel \chi \parallel :=|a_1|+\cdots +|a_n|,\mathrm{and}\mathrm{supp}\left(\chi \right):=\{x_1,\dots ,x_n\}.$$

From the definition of $L\left(X\right)$, a well-known fact easily follows: the dual space $L{\left(X\right)}^{\prime }$ of $L\left(X\right)$ is linearly isomorphic to the space $C\left(X\right)$ with pairing

$$\langle f,\chi \rangle =a_{1}f\left(x_1\right)+\cdots +a_{n}f\left(x_n\right)\mathrm{for}f\in C\left(X\right).$$

For every subset $A\subseteq L\left(X\right)$, we let $\mathrm{supp}\left(A\right):={\bigcup }_{\chi \in A}\mathrm{supp}\left(\chi \right)$ and $C_A:=sup\left(\{\parallel \chi \parallel :\chi \in A\}\cup \left\{0\right\}\right)$.

Theorem 6.

We let $1\le p\le q\le \infty$, and let $L\left(X\right)$ be the free locally convex space over a Tychonoff space X. Then,

(i)

each $(p,q)$-limited subset of $L\left(X\right)$ is finite-dimensional;

(ii)

$L\left(X\right)$ has the $G{P}_{(p,q)}$ property, the sequential $G{P}_{(p,q)}$ property, and the b-$G{P}_{(p,q)}$ property;

(iii)

every locally convex space E is a quotient space of $L\left(E\right)$.

Proof. 

(i) We suppose for a contradiction that there is a one-to-one infinite-dimensional sequence $A={\left\{{\chi }_n\right\}}_{n\in \omega }$ in $L\left(X\right)\setminus \left\{0\right\}$ which is a $(p,q)$-limited set. Since the support of any $\chi \in L\left(X\right)$ is finite and A is infinite-dimensional, without loss of generality, we can assume that A satisfies the following condition:

(a)

for every $n\in \omega$, there is $x_{n+1}\in \mathrm{supp}\left({\chi }_{n+1}\right)\setminus {\bigcup }_{i\le n}\mathrm{supp}\left({\chi }_i\right)$.

We fix an arbitrary $x_0\in \mathrm{supp}\left({\chi }_0\right)$. Passing to a subsequence of A if needed, we can also assume that there is sequence ${\left\{U_n\right\}}_{n\in \omega }$ of open subsets of X such that

(b)

$U_n\cap \mathrm{supp}\left({\chi }_n\right)=\left\{x_n\right\}$ for every $n\in \omega$;

(c)

$U_n\cap U_m=\varnothing$ for all distinct $n,m\in \omega$.

For every $n\in \omega$ and taking into account (b), we choose $f_n\in C\left(X\right)$ such that

(d)

$f_n\left(X\setminus U_n\right)=\left\{0\right\}$;

(e)

$\langle f_n,{\chi }_n\rangle =f_n\left(x_n\right)·{\chi }_n\left[x_n\right]=1$.

Since the support of any $\chi \in L\left(X\right)$ is finite, (c) and (d) imply that sequence ${\left\{f_n\right\}}_{n\in \omega }$ is weak* p-summable in $C\left(X\right)=L{\left(X\right)}^{\prime }$ for every $p\in [1,\infty ]$. On the other hand, (e) implies

$$\underset{\chi \in A}{\sup }|\langle f_n,\chi \rangle |\ge \underset{i\in \omega }{sup}\left|\langle f_n,{\chi }_i\rangle \right|\ge \langle f_n,{\chi }_n\rangle =1\nrightarrow 0,$$

which means that A is not a $(p,q)$-limited set, a contradiction.

(ii) immediately follows from (i).

(iii) By the definition of $L\left(E\right)$, the identity map ${\mathsf{id}}_E:E\to E$ can be extended to operator $I:L\left(E\right)\to E$. Since E is a closed subspace of $L\left(E\right)$, it is clear that I is a quotient map. □

For the coarse p-limited subsets of $L\left(X\right)$, the situation is antipodal to the $(p,q)$-limited subsets of $L\left(X\right)$.

Theorem 7.

We let $p\in [1,\infty ]$ and let $L\left(X\right)$ be the free locally convex space over a Tychonoff space X. Then,

(i)

each bounded subset B of $L\left(X\right)$ is coarse p-limited;

(ii)

$L\left(X\right)$ has the coarse $G{P}_p$ property if and only if each functionally bounded subset of X is finite.

Proof. 

(i) Recall that Proposition 2.7 of [41] states that subset A of $L\left(X\right)$ is bounded if and only if $\mathrm{supp}\left(A\right)$ is functionally bounded in X and $C_A$ is finite. We let B be a bounded subset of $L\left(X\right)$ and let $T\in \mathcal{L}\left(L\left(X\right),{\ell }_p\right)$ (or $T\in \mathcal{L}\left(L\left(X\right),c_0\right)$ if $p=\infty$). Then, $\mathrm{supp}\left(B\right)$ is functionally bounded in X, and hence $T\left(\mathrm{supp}\right(B\left)\right)$ is functionally bounded in the Banach space ${\ell }_p$ (or in $c_0$). Since any metric space is a $\mu$-space, the closure K of $T\left(\mathrm{supp}\right(B\left)\right)$ is compact. Now, inclusion $T\left(B\right)\subseteq C_B·\overline{\mathrm{acx}}\left(K\right)$ implies that $T\left(B\right)$ is a relatively compact subset of ${\ell }_p$ (or of $c_0$). Thus, B is a coarse p-limited subset of $L\left(X\right)$.

(ii) We assume that $L\left(X\right)$ has the coarse $G{P}_p$ property. Then, by (i), every bounded subset B of $L\left(X\right)$ has compact closure. In particular, $L\left(X\right)$ is quasi-complete. Therefore, by Theorem 3.8 of [27], each functionally bounded subset of X is finite. Conversely, if all functionally bounded subsets of X are finite, then, by Proposition 2.7 of [41], any bounded subset of $L\left(X\right)$ is finite-dimensional and hence relatively compact. Thus, $L\left(X\right)$ trivially has the coarse $G{P}_p$ property. □

Theorems 6 and 7 suggest the following problem.

Problem 7.

We let $1\le p\le q\le \infty$. We characterize Tychonoff spaces X for which $L\left(X\right)$ has one of the $V_{(p,q)}^{*}$ type properties defined in Definition 13.

We know from (ii) of Lemma 2 that the case $(1,\infty )$ is the strongest one in the sense that if E has some $G{P}_{(1,\infty )}$ type property, then it has the same $G{P}_{(p,q)}$ type property for all $1\le p\le q\le \infty$. This result and (v) of Proposition 6 motivate the problem of whether, for example, the $spG{P}_{(1,\infty )}$ property implies some addition conditions on space E. We answer this problem in the affirmative in the rest of this section.

Theorem 8.

We let E be a strict $\left(LF\right)$-space. If E does not contain an isomorphic copy of ${\ell }_1$ which is complemented in E, then the following assertions are equivalent:

(i)

E has the $G{P}_{(1,\infty )}$ property;

(ii)

E has the coarse $G{P}_1$ property;

(iii)

E is a Montel space.

If (i)–(iii) hold, then E is separable.

Proof. 

Since strict $\left(LF\right)$-spaces are barrelled by Proposition 11.3.1 of [16], (i) and (ii) are equivalent by (v) of Proposition 6. Corollary 5.16 of [27] states that a subset of E is bounded if and only if it is a coarse 1-limited set. Therefore, E has the coarse $G{P}_1$ property if and only if every bounded subset of E is relatively compact, i.e., E is a Montel space.

To prove that E is separable, we let $E:={\underrightarrow{s-\mathsf{ind}}}_n{E}_n$, where all $E_n$ are closed Fréchet subspaces of E. By Proposition 11.5.4(b) of [16] and (iii), all spaces $E_n$ are Montel. Therefore, by Theorem 11.6.2 of [16], all spaces $E_n$ are separable. Thus, $E={\bigcup }_{n\in \omega }{E}_n$ is also separable. □

Remark 5.

We let Γ be an uncountable set and let $1<p<\infty$. Then, by Theorem 8, the reflexive non-separable Banach space ${\ell }_p(\Gamma )$ has neither the $G{P}_{(1,\infty )}$ property nor the coarse $G{P}_1$ property. However, ${\ell }_p(\Gamma )$ has the coarse $G{P}_p$ property by Remark 3(2) of [20], and it has the Gelfand–Phillips property by Proposition 6.

Proposition 10.

We let E be a sequentially complete barrelled locally convex space. If E has the $spG{P}_{(1,\infty )}$ property, then E has the Schur property.

Proof. 

We let $S={\left\{x_n\right\}}_{n\in \omega }$ be a weakly null sequence in E. Then, S is weakly sequentially precompact. Therefore, by Corollary 3.14 of [27], S is a $(1,\infty )$-limited set. Hence, by the $spG{P}_{(1,\infty )}$ property, S is sequentially precompact. Thus, by Lemma 1, $x_n\to 0$ in E, and hence E has the Schur property. □

In general, the converse in Proposition 10 is not true as Example 1 shows. Nevertheless, in some important cases, the converse holds true.

Proposition 11.

We let E be a sequentially complete barrelled locally convex space whose bounded subsets are weakly sequentially precompact. Then, E has the $spG{P}_{(1,\infty )}$ property if and only if it has the Schur property.

Proof. 

The necessity follows from Proposition 10. To prove sufficiency, we assume that E has the Schur property and let A be a $(1,\infty )$-limited subset of E. Since A is bounded, it is weakly sequentially precompact. Then, by Lemma 2.2 of [27], A is sequentially precompact. Thus, E has $spG{P}_{(1,\infty )}$ property. □

Corollary 1.

We let E be a strict $\left(LF\right)$-space that does not contain an isomorphic copy of ${\ell }_1$. Then, E has the Schur property if and only if it has one (and hence all) of properties $G{P}_{(1,\infty )}$, $EG{P}_{(1,\infty )}$, $sG{P}_{(1,\infty )}$, $sEG{P}_{(1,\infty )}$, $prG{P}_{(1,\infty )}$, $spG{P}_{(1,\infty )}$, $prEG{P}_{(1,\infty )}$, $spEG{P}_{(1,\infty )}$, and b-$G{P}_{(1,\infty )}$.

Proof. 

By Proposition 4, it suffices to consider only the $spG{P}_{(1,\infty )}$ property. It is well known that E is a complete barrelled space. By the Ruess Theorem [29] mentioned before Theorem 4, E has the Rosenthal property. Since E has no an isomorphic copy of ${\ell }_1$, it follows that every bounded sequence has a weakly Cauchy subsequence (i.e., every bounded subset is weakly sequentially precompact). Now, Proposition 11 applies. □

4. Characterizations of Gelfand–Phillips Types Properties

In this section, we characterize locally convex spaces with Gelfand–Phillips type properties, in particular, in some important partial cases.

We let $p,q\in [1,\infty ]$ and let E and L be locally convex spaces. Generalizing the notions of limited, limited completely continuous, and limited p-convergent operators between Banach spaces introduced in [3,21,30], respectively, and following [42], a linear map $T:E\to L$ is called $(p,q)$-limited if $T\left(U\right)$ is a $(p,q)$-limited subset of L for some $U\in {\mathcal{N}}_0\left(E\right)$; if $p=q$ or $p=q=\infty$, we say that T is p-limited or limited, respectively.

Theorem 9.

We let $1\le p\le q\le \infty$ and let E be a locally convex space. Then, the following assertions are equivalent:

(i)

E has the $prG{P}_{(p,q)}$ (resp., $G{P}_{(p,q)}$, $sG{P}_{(p,q)}$, or $spG{P}_{(p,q)}$) property;

(ii)

for every locally convex space L, if operator $T:L\to E$ transforms bounded sets of L to $(p,q)$-limited sets of E, then T transforms bounded sets of L to precompact (resp., relatively compact, relatively sequentially compact, or sequentially precompact) subsets of E;

(iii)

every $(p,q)$-limited operator $T:L\to E$ from a locally convex space L to E is precompact (resp., compact, sequentially compact, or sequentially precompact);

(iv)

every $(p,q)$-limited operator $T:L\to E$ from a normed space L to E is precompact (resp., compact, sequentially compact, or sequentially precompact).

If E is sequentially complete, then (i)–(iv) are equivalent to the following

(v)

every $(p,q)$-limited operator $T:L\to E$ from a Banach space L to E is precompact (resp., compact, sequentially compact, or sequentially precompact).

Proof. 

(i)⇒(ii) We let $T:L\to E$ be an operator which transforms bounded sets of L to $(p,q)$-limited sets of E and let B be a bounded subset of L. By assumption, $T\left(B\right)$ is a $(p,q)$-limited subset of E. Then, by (i), $T\left(B\right)$ is a precompact (resp., relatively compact, relatively sequentially compact, or sequentially precompact) subset of E, as desired.

Implications (ii)⇒(iii)⇒(iv) are trivial.

(iv)⇒(i) We fix a $(p,q)$-limited subset B of E. It is clear that the closed absolutely convex hull D of the set B in E is also $(p,q)$-limited. We let L be the linear hull of D. Since D is bounded in E, function $\parallel ·\parallel :L\to [0,\infty )$, $\parallel ·\parallel :x\mapsto inf\{r\ge 0:x\in rD\}$ is a well-defined norm on linear space L, and set D coincides with the closed unit ball $B_L$ of normed space $(L,\parallel ·\parallel )$. Since identity inclusion $T:(L,\parallel ·\parallel )\to E$ is continuous and set $D=T\left(B_L\right)$ is $(p,q)$-limited in E, operator T is $(p,q)$-limited. By (iv), set $D=T\left(B_L\right)$ is precompact (resp., relatively compact, relatively sequentially compact, or sequentially precompact) in E, and hence so is set $B\subseteq D$. Thus, E has the $prG{P}_{(p,q)}$ (resp., $G{P}_{(p,q)}$, $sG{P}_{(p,q)}$, or $spG{P}_{(p,q)}$) property.

We assume that E is sequentially complete. Then, implication (iv)⇒(v) is trivial. To prove implication (v)⇒(iv), we let $T:L\to E$ be a $(p,q)$-limited operator from normed space L to E. Then, by Proposition 3.7 of [17], T can be extended to a bounded operator $\overline{T}$ from completion $\overline{L}$ of L to E. We observe that $\overline{T}\left(B_{\overline{L}}\right)\subseteq \overline{T\left(B_L\right)}$, and hence $\overline{T}$ is also $(p,q)$-limited. Thus, by (v), operator $\overline{T}$ and hence also T are precompact (resp., compact, sequentially compact, or sequentially precompact). □

We let $1\le p\le q\le \infty$ and let E and L be locally convex spaces. Following Definition 16.1 of [17], linear map $T:E\to L$ is called $(q,p)$-convergent if it sends weakly p-summable sequences in E to strongly q-summable sequences in L. We recall (see § 19.4 of [16]) that sequence ${\left\{x_n\right\}}_{n\in \omega }$ in an lcs L is called strongly p-summable if $\left(q_U\left(x_n\right)\right)\in {\ell }_p$ (or $\left(q_U\left(x_n\right)\right)\in c_0$ if $p=\infty$) for every $U\in {\mathcal{N}}_0^c\left(E\right)$, where, as usual, $q_U$ denotes the gauge functional of U.

Theorem 10.

We let $1\le p\le q\le \infty$ and let E be a locally convex space. Then, the following assertions are equivalent:

(i)

if L is a locally convex space and $T:L\to E$ is an operator such that $T^{*}:E_{w^{*}}^{\prime }\to L_{\beta }^{\prime }$ is $(q,p)$-convergent, then T transforms bounded sets of L to relatively sequentially compact (resp., sequentially precompact) subsets of E;

(ii)

if L is a normed space and $T:L\to E$ is an operator such that $T^{*}:E_{w^{*}}^{\prime }\to L_{\beta }^{\prime }$ is $(q,p)$-convergent, then T is sequentially compact (resp., sequentially precompact);

(iii)

the same as (ii) with $L={\ell }_1^0$;

(iv)

E has the $sG{P}_{(p,q)}$ property (resp., the $spG{P}_{(p,q)}$ property).

Moreover, if E is locally complete, then (i)–(iv) are equivalent to

(v)

the same as (ii) with $L={\ell }_1$.

Proof. 

Implications (i)⇒(ii)⇒(iii) and (ii)⇒(v) are trivial.

(iii)⇒(iv) and (v)⇒(iv): We let A be a $(p,q)$-limited subset of E. We fix an arbitrary sequence $S={\left\{x_n\right\}}_{n\in \omega }$ in A, so S is a bounded subset of E. Therefore, by Proposition 5.9 of [27], linear map $T:{\ell }_1^0\to E$ (or $T:{\ell }_1\to E$ if E is locally complete) defined by

$$T(a_0{e}_0+\cdots +a_n{e}_n):=a_0{x}_0+\cdots +a_n{x}_n(n\in \omega ,a_0,\dots ,a_n\in \mathbb{F}).$$

is continuous. For every $n\in \omega$ and each $\chi \in E^{\prime }$, we have $\langle T^{*}\left(\chi \right),e_n\rangle =\langle \chi ,T\left(e_n\right)\rangle =\langle \chi ,x_n\rangle$ and hence $T^{*}\left(\chi \right)={\left(\langle \chi ,x_n\rangle \right)}_n\in {\ell }_{\infty }$. In particular, $\parallel T^{*}{\left(\chi \right)\parallel }_{{\ell }_{\infty }}={sup}_{n\in \omega }\left|\langle \chi ,x_n\rangle \right|$.

We now let ${\left\{{\chi }_n\right\}}_{n\in \omega }$ be a weak* p-summable sequence in $E_{w^{*}}^{\prime }$. Since A and hence also S are $(p,q)$-limited sets, we obtain $\left(\parallel T^{*}\left({\chi }_n\right){\parallel }_{{\ell }_{\infty }}\right)=\left({sup}_{i\in \omega }\left|\langle {\chi }_n,x_i\rangle \right|\right)\in {\ell }_q$ (or $\in c_0$ if $q=\infty$). Therefore, $T^{*}$ is $(q,p)$-convergent, and hence, by (iii) or (v), operator T is sequentially compact (resp., sequentially precompact). Therefore, $S={\left\{T\left(e_n\right)\right\}}_{n\in \omega }$ has a convergent (resp., Cauchy) subsequence in E. Hence, A is a relatively sequentially compact (resp., sequentially precompact) subset of E. Thus, E has the $sG{P}_{(p,q)}$ property (resp., the $spG{P}_{(p,q)}$ property).

(iv)⇒(i) We let $T:L\to E$ be an operator from a locally convex space $(L,\tau )$ such that $T^{*}:E_{w^{*}}^{\prime }\to L_{\beta }^{\prime }$ is $(q,p)$-convergent. We fix an arbitrary bounded subset B of L. Then, $U:=B^{\circ }$ is a neighbourhood of zero of $L_{\beta }^{\prime }$. We let ${\left\{{\chi }_n\right\}}_{n\in \omega }$ be a weakly p-summable sequence in $E_{w^{*}}^{\prime }$. By assumption, $T^{*}:E_{w^{*}}^{\prime }\to L_{\beta }^{\prime }$ is $(q,p)$-convergent, and hence

$$\left(q_U\left(T^{*}\left({\chi }_n\right)\right)\right)\in {\ell }_q(\mathrm{or}\in c_0\mathrm{if}q=\infty ),$$

(2)

where $q_U$ is the gauge functional of U. For every $\chi \in E^{\prime }$, we have

$$\begin{array}{cc}\hfill q_U\left(T^{*}\left(\chi \right)\right)& =inf\left\{\lambda >0:T^{*}\left(\chi \right)\in \lambda B^{\circ }\right\}=inf\{\lambda >0:\underset{x\in B}{sup}|\langle T^{*}\left(\chi \right),x\rangle |\le \lambda \}\hfill \\ & =\underset{x\in B}{sup}|\langle T^{*}\left(\chi \right),x\rangle |=\underset{x\in B}{sup}\left|\langle \chi ,T\left(x\right)\rangle \right|.\hfill \end{array}$$

Therefore, (2) means that set $T\left(B\right)$ is a $(p,q)$-limited subset of E. By (iv), we determine that $T\left(B\right)$ is a relatively sequentially compact (resp., sequentially precompact) subset of E. □

We let $p,q,q^{\prime }\in [1,\infty ]$, $q^{\prime }\le q$ and let E and L be locally convex spaces. Following [42], linear map $T:E\to L$ is called $(q^{\prime },q)$-limited p-convergent if $T\left(x_n\right)\to 0$ for every weakly p-summable sequence ${\left\{x_n\right\}}_{n\in \omega }$ in E which is a $(q^{\prime },q)$-limited subset of E.

Our next aim is to offer another characterization of locally convex spaces E with the sequentially precompact $G{P}_{(q^{\prime },q)}$ property under an additional assumption that E has $wCSP$. First, we prove the following lemma.

Lemma 6.

We let $1\le p\le q\le \infty$ and let E be a locally convex space. If E has the $spG{P}_{(p,q)}$ property, then the identity map ${\mathsf{id}}_E:E\to E$ is $(p,q)$-limited ∞-convergent.

Proof. 

We fix a $(p,q)$-limited weakly null sequence $S={\left\{x_n\right\}}_{n\in \omega }$ in E and suppose for a contradiction that $x_n\nrightarrow 0$ in E. Passing to a subsequence, if needed, we can assume that $S\cap U=\varnothing$ for some neighborhood U of zero in E. Since S is a $(p,q)$-limited set and E has the $spG{P}_{(p,q)}$ property, S is sequentially precompact and hence precompact in E. As S is also weakly null, Lemma 1, implies that $x_n\to 0$. But this is impossible since $S\cap U=\varnothing$. □

Theorem 11.

We let $1\le p\le q\le \infty$ and let E be a locally convex space with $wCSP$. Then, E has the $spG{P}_{(p,q)}$ property if and only if the identity map ${\mathsf{id}}_E:E\to E$ is $(p,q)$-limited ∞-convergent.

Proof. 

The necessity immediately follows from Lemma 6. To prove sufficiency, we assume that identity map ${\mathsf{id}}_E:E\to E$ is $(p,q)$-limited ∞-convergent and let A be a $(p,q)$-limited subset of E. We have to show that any sequence $S={\left\{x_n\right\}}_{n\in \omega }$ in A contains a Cauchy subsequence (in E). Since E has $wCSP$, S has subsequence $S^{\prime }={\left\{x_{n_k}\right\}}_{k\in \omega }$ which is weakly Cauchy. We let ${\left\{k_i\right\}}_{i\in \omega }$ be a strictly increasing sequence in $\omega$. Then, ${\{x_{n_{k_i}}-x_{n_{k_{i+1}}}\}}_{k\in \omega }$ is weakly null. As ${\{x_{n_{k_i}}-x_{n_{k_{i+1}}}\}}_{k\in \omega }$ is also a $(p,q)$-limited set and ${\mathsf{id}}_E$ is $(p,q)$-limited ∞-convergent, it follows that $x_{n_{k_i}}-x_{n_{k_{i+1}}}\to 0$ in E. Thus, $S^{\prime }$ is Cauchy in E. □

We know from Theorem 1 that a Banach space E has the Gelfand–Phillips property if and only if every limited weakly null sequence in E is norm null. An analogue assertion for the coarse $G{P}_p$ property (and $2\le p<\infty$) was obtained in Theorem 3 of [20]. Below, we generalize both these results.

Theorem 12.

We let $1\le p\le q\le \infty$ and let $(E,\tau )$ be a locally convex space.

(i)

If E has the $prG{P}_{(p,q)}$ property (the $spG{P}_{(p,q)}$ property), then every weakly null $(p,q)$-limited sequence in E is τ-null.

(ii)

If E has the coarse $prG{P}_p$ property (the coarse $spG{P}_p$ property), then every weakly null coarse p-limited sequence in E is τ-null.

(iii)

We assume, additionally, that $p\ge 2$ and E have the Rosenthal property. Then, E has the $prG{P}_{(p,q)}$ property if and only if every weakly null $(p,q)$-limited sequence in E is τ-null.

(iv)

We assume, additionally, that $p\ge 2$ and E have the Rosenthal property. Then, E has the coarse $prG{P}_p$ property if and only if every weakly null coarse p-limited sequence in E is τ-null.

Proof. 

(i) and (ii): We let $S={\left\{x_n\right\}}_{n\in \omega }$ be a weakly null $(p,q)$-limited (resp., coarse p-limited) sequence in E. By the $prG{P}_{(p,q)}$ property (resp., the $spG{P}_{(p,q)}$ property, the coarse $prG{P}_p$ property, or the coarse $spG{P}_p$ property), S is a precompact subset of E. Thus, by Lemma 1, S is $\tau$-null.

(iii) and (iv): We assume that $p\ge 2$ and E have the Rosenthal property. The necessity is proven in (i) and (ii). To prove sufficiency, we let every weakly null $(p,q)$-limited (resp., coarse p-limited) sequence in E to be $\tau$-null. We let A be a $(p,q)$-limited (resp., coarse p-limited) subset of E and suppose for a contradiction that A is not $\tau$-precompact. Then, there are sequence ${\left\{a_n\right\}}_{n\in \omega }$ in A and $U\in {\mathcal{N}}_0\left(E\right)$ such that $a_n-a_m\notin U$ for all distinct $n,m\in \omega$. By Theorem 5.21 of [27] and Theorem 4, we determine that A is weakly sequentially precompact. Therefore, passing to a subsequence, if needed, we can assume that ${\left\{a_n\right\}}_{n\in \omega }$ is weakly Cauchy. In particular, sequence ${\{a_n-a_{n+1}\}}_{n\in \omega }$ is weakly null. Since ${\{a_n-a_{n+1}\}}_{n\in \omega }$ is also a $(p,q)$-limited (resp., coarse p-limited) subset of E (see Lemmas 3.1 and 4.1 of [27]), it follows that $a_n-a_{n+1}\to 0$ in E, which contradicts condition $a_n-a_{n+1}\notin U$ for all $n\in \omega$. Thus, A is $\tau$-precompact, and hence E has the $prG{P}_{(p,q)}$ property (resp., the coarse $prG{P}_p$ property). □

We let D be a subset of a locally convex space E. Recall that subset A of E is called D-separated if $a-a^{\prime }\notin D$ for all distinct $a,a^{\prime }\in A$, and A is separated if A is U-separated for some neighborhood U of zero. We also denote by ${\left({\ell }_r\right)}_p$ and ${\left(c_0\right)}_p$ the Banach spaces ${\ell }_r$ and $c_0$ endowed with the topology induced from ${\mathbb{F}}^{\omega }$.

Since any limited set and each $(p,q)$-limited set are $(p,\infty )$-limited, case $q=\infty$ is of interest. Below, we offer necessary conditions to have the precompact $G{P}_{(p,\infty )}$ property.

Proposition 12.

We let $p\in [1,\infty ]$ and let E be a locally convex space. We consider the following assertions:

(i)

E has the precompact $G{P}_{(p,\infty )}$ property;

(ii)

For every bounded non-precompact set $B\subseteq E$, there is a weak* p-summable sequence ${\left\{{\chi }_n\right\}}_{n\in \omega }$ in $E^{\prime }$ such that $\parallel {\chi }_n{\parallel }_B\nrightarrow 0$.

(iii)

For any infinite bounded separated subset D of E and every $\delta >0$, there exist a sequence ${\left\{x_n\right\}}_{n\in \omega }$ in D and a weak* p-summable sequence ${\left\{f_n\right\}}_{n\in \omega }$ in $E^{\prime }$ such that $|f_n\left(x_k\right)|<\delta$ and $|f_n\left(x_n\right)|>1+\delta$ for all natural numbers $k<n$.

(iv)

For any infinite bounded separated set D in E, there exists a continuous operator $T:E\to {\left({\ell }_p\right)}_p$ (or $T:E\to {\left(c_0\right)}_p$ if $p=\infty$) such that $T\left(D\right)$ is not precompact in the Banach space ${\ell }_p$ (or in $c_0$ if $p=\infty$).

Then, (i)⇒(ii)⇒(iii)⇒(iv).

Proof. 

(i)⇒(ii) We fix any bounded non-precompact set $B\subseteq E$. By (i), E has the precompact $G{P}_{(p,\infty )}$ property and hence B is not $(p,\infty )$-limited. Therefore, there exists a weak* p-summable sequence ${\left\{{\chi }_n\right\}}_{n\in \omega }$ in $E^{\prime }$ such that $\parallel {\chi }_n{\parallel }_B\nrightarrow 0$, as desired.

(ii)⇒(iii) We fix any $\delta >0$ and any infinite bounded separated set D in E. We choose $U\in {\mathcal{N}}_0\left(E\right)$ such that the set D is U-separated. We observe that D is not precompact because D is U-separated and infinite. By (ii), there is a weak* p-summable sequence ${\left\{g_n\right\}}_{n\in \omega }$ in $E^{\prime }$ such that $\parallel g_n{\parallel }_D\nrightarrow 0$. Passing to a subsequence, if needed, we can assume that $\parallel g_n{\parallel }_D>a>0$ for all $n\in \omega$. Further, multiplying each functional $g_n$ by $(1+\delta )/a$, we can assume that

$$\parallel g_n{\parallel }_D=\underset{d\in D}{sup}\left|g_n\left(d\right)\right|>1+\delta \mathrm{for}\mathrm{every}n\in \omega .$$

(3)

Now, we choose an arbitrary $x_0\in D$ such that $|g_0\left(x_0\right)|>1+\delta$. We assume that, for $k\in \omega$, we find $x_0,\dots ,x_k\in D$ and sequence $0=n_0<n_1<\cdots <n_k$ of natural numbers such that

$$|g_{n_j}\left(x_i\right)|<\delta \mathrm{and}|g_{n_j}\left(x_j\right)|>1+\delta \mathrm{for}\mathrm{every}0\le i<j\le k.$$

Since $g_n\to 0$ in the weak* topology $\sigma (E^{\prime },E)$, (3) implies that there are $n_{k+1}>n_k$ and $x_{k+1}\in D\setminus \{x_0,\dots ,x_k\}$ such that $|g_{n_{k+1}}\left(x_{k+1}\right)|>1+\delta$ and $|g_{n_{k+1}}\left(x_i\right)|<\delta$ for every $i\le k$. For every $k\in \omega$, we put $f_k:=g_{n_k}$, and observe that subsequence ${\left\{f_k\right\}}_{k\in \omega }$ of ${\left\{g_n\right\}}_{n\in \omega }$ is also weak* p-summable in $E^{\prime }$ and

$$|f_k\left(x_i\right)|=|g_{n_k}\left(x_i\right)|<\delta \mathrm{and}|f_k\left(x_k\right)|=|g_{n_k}\left(x_k\right)|>1+\delta$$

for any numbers $i<k$, as desired.

(iii)⇒(iv) We let D be any bounded separated set in E. By (iii) applied for $\delta =\frac{1}{2}$, there exist a sequence ${\left\{x_n\right\}}_{n\in \omega }$ in D and a weak* p-summable sequence ${\left\{{\chi }_n\right\}}_{n\in \omega }$ in $E^{\prime }$ such that $|{\chi }_m\left(x_n\right)|<{\displaystyle \frac{1}{2}}$ and $|{\chi }_m\left(x_m\right)|>{\displaystyle \frac{3}{2}}$ for all $n<m$. Then, $|{\chi }_m\left(x_m\right)-{\chi }_m\left(x_n\right)|\ge |{\chi }_m\left(x_m\right)|-|{\chi }_m\left(x_n\right)|>{\displaystyle \frac{3}{2}}-{\displaystyle \frac{1}{2}}=1$ for every $n<m$. It follows that the operator

$$T:E\to {\left({\ell }_p\right)}_p(orT:E\to {\left(c_0\right)}_{p}ifp=\infty ),T\left(x\right):={\left({\chi }_n\left(x\right)\right)}_{n\in \omega },$$

is well-defined and continuous. Since

$$\parallel T\left(x_n\right)-T\left(x_m\right)\parallel \ge |{\chi }_m\left(x_m\right)-{\chi }_m\left(x_n\right)|>1\mathrm{for}\mathrm{every}n<m,$$

the set $T\left(D\right)\supseteq {\left\{T\left(x_n\right)\right\}}_{n\in \omega }$ is not precompact in ${\ell }_p$ (or in $c_0$ if $p=\infty$). □

In two extreme cases when $p=1$ or $p=\infty$, we can reverse implications in Proposition 12, cf. Theorem 2.2 of [13].

Theorem 13.

We let $p\in \{1,\infty \}$. For locally convex space E, the following assertions are equivalent:

(i)

E has the precompact $G{P}_{(p,\infty )}$ property;

(ii)

For every bounded non-precompact set $B\subseteq E$, there is a weak* p-summable sequence ${\left\{{\chi }_n\right\}}_{n\in \omega }$ in $E^{\prime }$ such that $\parallel {\chi }_n{\parallel }_B\nrightarrow 0$.

(iii)

For any infinite bounded separated subset D of E and every $\delta >0$, there exist a sequence ${\left\{x_n\right\}}_{n\in \omega }$ in D and a weak* p-summable sequence ${\left\{f_n\right\}}_{n\in \omega }$ in $E^{\prime }$ such that $|f_n\left(x_k\right)|<\delta$ and $|f_n\left(x_n\right)|>1+\delta$ for all natural numbers $k<n$.

(iv)

For any infinite bounded separated set D in E, there exists a continuous operator $T:E\to {\left({\ell }_1\right)}_p$ (or $T:E\to {\left(c_0\right)}_p$ if $p=\infty$) such that $T\left(D\right)$ is not precompact in the Banach space ${\ell }_1$ (or in $c_0$ if $p=\infty$).

Proof. 

Implications (i)⇒(ii)⇒(iii)⇒(iv) are proven in Proposition 12.

(iv)⇒(i) To show that E has the precompact $G{P}_{(p,\infty )}$ property, we show that any non-precompact subset of E is not $(p,\infty )$-limited. We fix a bounded non-precompact subset $P\subseteq E$. Then, there exists $U\in {\mathcal{N}}_0\left(E\right)$ such that $L⊈F+U$ for any finite subset $F\subseteq E$. For every $n\in \omega$, we inductively choose point $z_n\in P$ so that $z_n\notin {\bigcup }_{k<n}(z_k+U)$. We observe that set $D:=\{z_n:n\in \omega \}$ is infinite, bounded and U-separated, so it is non-precompact. By (iv), there exists a continuous operator $T:E\to {\left({\ell }_1\right)}_p$ (or $T:E\to {\left(c_0\right)}_p$ if $p=\infty$) such that set $T\left(D\right)$ is not precompact in the Banach space ${\ell }_1$ (or in $c_0$ if $p=\infty$). We distinguish between two cases.

Case 1. We assume that $p=\infty$. We follow the idea of Theorem 2.2 of [13]. Since $c_0$ is a Banach space and $T\left(D\right)$ is not precompact, there exist a sequence ${\left\{a_n\right\}}_{n\in \omega }$ in D and $\delta >0$ such that $\parallel T\left(a_n\right)-T\left(a_m\right){\parallel }_{c_0}\ge \delta$ for all distinct $n,m\in \omega$ (see also (ii) of Proposition 1). We observe that sequence ${\left\{T\left(a_n\right)\right\}}_{n\in \omega }$ is bounded in Banach space $c_0$. Therefore, there are two sequences, $0\le n_0<n_1<\cdots$ and $0\le m_0<m_1<\cdots$, of natural numbers such that

$$\left|\langle e_{m_k}^{\prime },T\left(a_{n_k}\right)\rangle \right|>\frac{\delta }{2}\mathrm{for}\mathrm{every}k\in \omega ,$$

where $e_n^{\prime }:{\left(c_0\right)}_p\to \mathbb{F}$ is the nth coordinate functional. For every $k\in \omega$, we set $f_k:=e_{m_k}^{\prime }\circ T$. It follows that ${\left\{f_k\right\}}_{k\in \omega }$ is a weak* null in $E^{\prime }$ and

$$\parallel f_k{\parallel }_P=\underset{x\in P}{sup}|f_k\left(x\right)|\ge |f_k\left(a_{n_k}\right)|>\frac{\delta }{2}$$

for every $k\in \omega$, witnessing that set P is not limited.

Case 2. We assume that $p=1$. Recall that, by (i) of Proposition 1, a bounded subset A of ${\ell }_1$ is precompact if and only if

$$\underset{m\to \infty }{lim}sup\left\{\sum _{m\le n}\left|x_n\right|:x=\left(x_n\right)\in A\right\}=0.$$

(4)

For every $k\in \omega$, we set $y_k:=T\left(z_k\right)={\left(a_{n,k}\right)}_{n\in \omega }\in {\ell }_1$. Since $T\left(D\right)$ is non-precompact, (4) implies that there is $\epsilon >0$ such that

$$sup\left\{\sum _{m\le n}\left|a_{n,k}\right|:y_k=\left(a_{n,k}\right)\in T\left(D\right)\right\}>10\epsilon \mathrm{for}\mathrm{every}m\in \omega .$$

(5)

We let $i=0$. We set $m_0:=0$. By (5), we choose $k_0,r_0\in \omega$ such that $m_0<r_0$ and

$$\sum _{n=m_0}^{r_0}\left|a_{n,k_0}\right|>8\epsilon .$$

(6)

We choose $m_1>r_0$ such that

$$\sum _{n\ge m_1}\left|a_{n,k}\right|<\epsilon \mathrm{for}\mathrm{every}k\le k_0.$$

(7)

For $i=1$, (5)–(7) imply that there are $k_1,r_1\in \omega$ such that $k_1>k_0$, $m_1<r_1$ and

$$\sum _{n=m_1}^{r_1}\left|a_{n,k_1}\right|>8\epsilon .$$

(8)

We choose $m_2>r_1$ such that

$$\sum _{n\ge m_2}\left|a_{n,k}\right|<\epsilon \mathrm{for}\mathrm{every}k\le k_1.$$

Continuing this process, we find sequences $0=m_0<r_0<m_1<r_1<\cdots$ and $k_0<k_1<\cdots$ such that for every $i\in \omega$, we have

$$\sum _{n=m_i}^{r_i}|a_{n,k_i}|>8\epsilon \mathrm{and}\sum _{n\ge m_{i+1}}\left|a_{n,k_i}\right|<\epsilon .$$

(9)

For every $i\in \omega$, by Lemma 6.3 of [43], there is a subset $F_i$ of $[m_i,r_i]$ such that

$$\left|\sum _{n\in F_i}{a}_{n,k_i}\right|>2\epsilon .$$

(10)

For every $n\in \omega$, we let $e_n^{*}$ be the nth coordinate functional of the dual space ${\left({\ell }_1\right)}_p^{\prime }$ of ${\left({\ell }_1\right)}_p$. For every $i\in \omega$, we set

$${\chi }_i:=\sum _{n\in F_i}{e}_n^{*}.$$

We observe that sequence ${\left\{{\chi }_i\right\}}_{i\in \omega }$ is weak* 1-summable in ${\left({\ell }_1\right)}_p^{\prime }$ because if $y=\left(a_n\right)\in {\ell }_1$, then the inequalities $m_i<r_i<m_{i+1}$ imply that all $F_i$ are pairwise disjoint and hence

$$\sum _{i\in \omega }|\langle {\chi }_i,y\rangle |=\sum _{i\in \omega }|\sum _{n\in F_i}{a}_n|\le \sum _{i\in \omega }\sum _{n\in F_i}|a_n|\le \sum _{n\in \omega }|a_n{|=\parallel y\parallel }_{{\ell }_1}<\infty .$$

Therefore, sequence ${\left\{T^{*}\left({\chi }_i\right)\right\}}_{i\in \omega }$ is weak* 1-summable in $E^{\prime }$. For every $i\in \omega$, inequality (10) implies

$$\underset{x\in P}{sup}|\langle T^{*}\left({\chi }_i\right),x\rangle |\ge |\langle {\chi }_i,T\left(z_{k_i}\right)\rangle |=|\langle {\chi }_i,y_{k_i}\rangle |=|\sum _{n\in F_i}{a}_{n,k_i}|>2\epsilon ,$$

which means that set P is not $(1,\infty )$-limited. □

The next theorem immediately follows from Corollary 3.18 of [27] and, under some conditions, it characterizes the (precompact) $G{P}_{(p,\infty )}$ property.

Theorem 14.

We let E be a p-barrelled Mackey space such that $E_{w^{*}}^{\prime }$ is a weakly p-angelic space. Then, E has the precompact $G{P}_{(p,\infty )}$ property. If, in addition, E is von Neumann complete, then E has the $G{P}_{(p,\infty )}$ property.

Since case $p=\infty$ is of independent interest, we select the next corollary which follows from Corollary 3.19 of [27] (and in fact from Theorem 14).

Corollary 2.

We let a locally convex space E satisfy one of the following conditions:

(i)

E is a $c_0$-barrelled Mackey space such that $E_{w^{*}}^{\prime }$ is a weakly angelic space;

(ii)

E is a reflexive space such that $E_{\beta }^{\prime }$ is a weakly angelic space;

(iii)

E is a separable $c_0$-barrelled Mackey space.

Then, E has the precompact $GP$ property. If, in addition, E is von Neumann complete, then E has the $GP$ property.

We let $p\in [1,\infty ]$ and let E and L be locally convex spaces. Following [42], operator $T\in \mathcal{L}(E,L)$ is called coarse p-limited if there is $U\in {\mathcal{N}}_0\left(E\right)$ such that $T\left(U\right)$ is a coarse p-limited subset of L.

Banach spaces with the coarse p-Gelfand–Phillips property are characterized in Proposition 13 of [20]. The next theorem generalizes that result (we use in the proof the easy fact that subset A of E is precompact if and only if each sequence in A is precompact).

Theorem 15.

For $p\in [1,\infty ]$ and a locally convex space E, the following assertions are equivalent:

(i)

E has the coarse $sG{P}_p$ property (resp., the coarse $spG{P}_p$ property or the coarse $prG{P}_p$ property);

(ii)

for every locally convex space Y, if operator $T:Y\to E$ transforms bounded sets of Y to coarse p-limited sets of E, then T transforms bounded sets of Y to relatively sequentially compact (resp., sequentially precompact or precompact) subsets of E;

(iii)

for every normed space Y, each coarse p-limited operator $T:Y\to E$ is sequentially compact (resp., sequentially precompact or precompact);

(iv)

each coarse p-limited operator $T:{\ell }_1^0\to E$ is sequentially compact (resp., sequentially precompact or precompact).

If, in addition, E is locally complete, then (i)–(iv) are equivalent to

(v)

each coarse p-limited operator $T:{\ell }_1\to E$ is sequentially compact (resp., sequentially precompact or precompact).

Proof. 

Implications (i)⇒(ii)⇒(iii)⇒(iv) and (iii)⇒(v) are trivial.

(iv)⇒(i) and (v)⇒(i): We let A be a coarse p-limited subset of E. To show that A is relatively sequentially compact (resp., sequentially precompact or precompact), we let ${\left\{x_n\right\}}_{n\in \omega }$ be a sequence in A. Then, by Proposition 14.9 of [17], the linear map $T:{\ell }_1^0\to E$ (or $T:{\ell }_1\to E$ if E is locally complete) defined by

$$T(a_0{e}_0+\cdots +a_n{e}_n):=a_0{x}_0+\cdots +a_n{x}_n(n\in \omega ,a_0,\dots ,a_n\in \mathbb{F}).$$

is continuous. It is clear that $T\left(B_{{\ell }_1^0}\right)\subseteq \overline{\mathrm{acx}}\left({\left\{x_n\right\}}_{n\in \omega }\right)$ (or $T\left(B_{{\ell }_1}\right)\subseteq \overline{\mathrm{acx}}\left({\left\{x_n\right\}}_{n\in \omega }\right)$ if E is locally complete). Since $\overline{\mathrm{acx}}\left({\left\{x_n\right\}}_{n\in \omega }\right)\subseteq \overline{\mathrm{acx}}\left(A\right)$ is a coarse p-limited set (see (ii) of Lemma 4.1 of [27]), it follows that operator T is coarse p-limited. Therefore, by (iv) or (v), sequence ${\left\{x_n\right\}}_{n\in \omega }\subseteq T\left(B_{{\ell }_1^0}\right)$ has a convergent subsequence (resp., a Cauchy subsequence or $T\left({\left\{e_n\right\}}_{n\in \omega }\right)={\left\{x_n\right\}}_{n\in \omega }$ is precompact). Thus, A is relatively sequentially compact (resp., sequentially precompact, or precompact), as desired. □

5. p-Gelfand–Phillips Sequentially Compact Property of Order (q′, q)

We start this section with the next assertion. Recall (see Definition 10) that an lcs $(E,\tau )$ is said to have p-$GPsc{P}_{(q^{\prime },q)}$ if every weakly p-summable sequence in E which is also a $(q^{\prime },q)$-limited set is $\tau$-null.

Proposition 13.

We let $1\le q^{\prime }\le q\le \infty$ and let E be a locally convex space.

(i)

If E has $prG{P}_{(q^{\prime },q)}$, then it has p-$GPsc{P}_{(q^{\prime },q)}$ for every $p\in [1,\infty ]$.

(ii)

If $q^{\prime }\ge 2$ and E has the Rosenthal property, then E has $prG{P}_{(q^{\prime },q)}$ if and only if it has $GPsc{P}_{(q^{\prime },q)}$.

Proof. 

(i) follows from (i) of Theorem 12 and (ii) follows from (iii) of Theorem 12. □

Remark 6.

The converse in (i) of Proposition 13 is not true in general. Indeed, we let $1<r,p<2$ and $E={\ell }_r$. By Corollary 1, ${\ell }_r$ does not have $prG{P}_{(1,\infty )}$. On the other hand, we let ${\left\{x_n\right\}}_{n\in \omega }$ be a weakly p-summable sequence in E which is also a $(q^{\prime },q)$-limited set, and we suppose for a contradiction that $x_n\nrightarrow 0$. Passing to a subsequence, we can assume that $1/c\le \parallel x_n\parallel \le c$ for some $c>1$ and all $n\in \omega$. Dividing $x_n$ by its norm, we can assume also that ${\left\{x_n\right\}}_{n\in \omega }$ is a normalized sequence. Taking into account that ${\left\{x_n\right\}}_{n\in \omega }$ is weakly null, Proposition 2.1.3 of [44] implies that there is a subsequence ${\left\{x_{n_k}\right\}}_{k\in \omega }$ which is a basic sequence equivalent to the canonical basis of ${\ell }_r$ and such that $\overline{\mathsf{span}}\left({\left\{x_{n_k}\right\}}_{k\in \omega }\right)$ is complemented in ${\ell }_r$. It follows that the canonical basis ${\left\{e_n\right\}}_{n\in \omega }$ of ${\ell }_r$ is weakly p-summable, which contradicts Example 4.4 of [17] since $p<r^{*}$. Thus, $x_n\to 0$ and ${\ell }_r$ have p-$GPsc{P}_{(q^{\prime },q)}$ for all $1\le q^{\prime }\le q\le \infty$.□

We let $p,q\in [1,\infty ]$. Following [32], a locally convex space E is said to have the sequential Dunford–Pettis property of order $(p,q)$ (the sequential $D{P}_{(p,q)}$ property) if ${lim}_{n\to \infty }\langle {\chi }_n,x_n\rangle =0$ for every weakly p-summable sequence ${\left\{x_n\right\}}_{n\in \omega }$ in E and each weakly q-summable sequence ${\left\{{\chi }_n\right\}}_{n\in \omega }$ in $E_{\beta }^{\prime }$. Following [42], a linear map T from E to a locally convex space L is called weakly $(p,q)$-convergent if ${lim}_{n\to \infty }\langle {\eta }_n,T\left(x_n\right)\rangle =0$ for every weakly q-summable sequence ${\left\{{\eta }_n\right\}}_{n\in \omega }$ in $L_{\beta }^{\prime }$ and each weakly p-summable sequence ${\left\{x_n\right\}}_{n\in \omega }$ in E. The following proposition, which is used below, shows that the sequential $D{P}_{(p,q)}$ property of the range implies some strong additional properties of operators.

Proposition 14.

We let $p,q\in [1,\infty ]$ and let E and L be locally convex spaces. If L has the sequential $D{P}_{(p,q)}$ property (for instance, L is a quasibarrelled locally complete space with the Dunford–Pettis property, e.g., $L={\ell }_{\infty }$), then each operator $T\in \mathcal{L}(E,L)$ is weakly $(p,q)$-convergent.

Proof. 

We let ${\left\{{\eta }_n\right\}}_{n\in \omega }$ be a weakly q-summable sequence in $L_{\beta }^{\prime }$ and let ${\left\{x_n\right\}}_{n\in \omega }$ be a weakly p-summable sequence in E. Then, ${\left\{T\left(x_n\right)\right\}}_{n\in \omega }$ is a weakly p-summable sequence in L. Now, the sequential $D{P}_{(p,q)}$ property of L implies $\langle {\eta }_n,T\left(x_n\right)\rangle \to 0$. Thus, T is weakly $(p,q)$-convergent.

If L is a quasibarrelled locally complete space with the Dunford–Pettis property, Corollary 5.13 of [32] implies that L has the sequential $D{P}_{(p,q)}$ property. □

The following theorem characterizes locally convex spaces E with p-$GPsc{P}_{(q^{\prime },q)}$ and explains our usage of “sequentially compact” in the notion of p-$GPsc{P}_{(q^{\prime },q)}$.

Theorem 16.

We let $p,q,q^{\prime }\in [1,\infty ]$, $q^{\prime }\le q$. Then, for a locally convex space $(E,\tau )$, the following assertions are equivalent:

(i)

E has the p-$GPsc{P}_{(q^{\prime },q)}$;

(ii)

each operator $T\in \mathcal{L}(E,{\ell }_{\infty })$ is $(q^{\prime },q)$-limited p-convergent;

(iii)

each weakly $(p,q)$-convergent operator $T\in \mathcal{L}(E,{\ell }_{\infty })$ is $(q^{\prime },q)$-limited p-convergent;

(iv)

each weakly sequentially p-compact subset A of E which is a $(q^{\prime },q)$-limited set is relatively sequentially compact in E;

(v)

each weakly sequentially p-precompact subset A of E which is a $(q^{\prime },q)$-limited set is sequentially precompact in E.

Proof. 

(i)⇒(ii) We assume that E has p-$GPsc{P}_{(q^{\prime },q)}$ and let ${\left\{x_n\right\}}_{n\in \omega }$ be weakly p-summable sequence in E which is a $(q^{\prime },q)$-limited set. By p-$GPsc{P}_{(q^{\prime },q)}$, we have $x_n\to 0$ in E. Therefore, $T\left(x_n\right)\to 0$ is also in ${\ell }_{\infty }$. Thus, T is a $(q^{\prime },q)$-limited p-convergent operator.

(ii)⇒(i) We assume that each operator $T\in \mathcal{L}(E,{\ell }_{\infty })$ is $(q^{\prime },q)$-limited p-convergent. To show that E has p-$GPsc{P}_{(q^{\prime },q)}$, we let ${\left\{x_n\right\}}_{n\in \omega }$ be a weakly p-summable sequence in E which is a $(q^{\prime },q)$-limited set. Assuming that $x_n\nrightarrow 0$ in E and passing to a subsequence, if needed, we can assume that there is $U\in {\mathcal{N}}_0^c\left(E\right)$ such that $x_n\notin U$ for all $n\in \omega$. For every $n\in \omega$, we choose ${\chi }_n\in U^{\circ }$ such that $\langle {\chi }_n,x_n\rangle >1$. We define linear map $T:E\to {\ell }_{\infty }$ by

$$T\left(x\right):={\left(\langle {\chi }_n,x\rangle \right)}_n(x\in E).$$

Since $T\left(U\right)\subseteq B_{{\ell }_{\infty }}$, map T is continuous. As $\parallel T\left(x_n\right){\parallel }_{{\ell }_{\infty }}\ge \langle {\chi }_n,x_n\rangle >1\nrightarrow 0$, it follows that T is not $(q^{\prime },q)$-limited p-convergent. This is a contradiction.

Equivalence (ii)⇔(iii) immediately follows from Proposition 14.

(i)⇒(iv) To show that A is relatively sequentially compact in E, we take an arbitrary sequence $S={\left\{a_n\right\}}_{n\in \omega }$ in A. Since A is weakly sequentially p-compact, passing to a subsequence, we can assume that S weakly p-converges to some point $a\in A$. So sequence ${\{a_n-a\}}_{n\in \omega }$ is weakly p-summable and also a $(q^{\prime },q)$-limited set. Therefore, by p-$GPsc{P}_{(q^{\prime },q)}$, we have $a_n-a\to 0$. Thus, A is a relatively sequentially compact subset of E.

(iv)⇒(i) If $S={\left\{a_n\right\}}_{n\in \omega }$ is a weakly p-summable sequence in E which is a $(q^{\prime },q)$-limited set, then S is relatively sequentially compact in E. Being also weakly null, S is a null sequence in E by Lemma 1.

(i)⇒(v) To show that A is sequentially precompact in E, we take an arbitrary sequence $S={\left\{a_n\right\}}_{n\in \omega }$ in A. Since A is weakly sequentially p-precompact, we can assume that S is weakly p-Cauchy. For every strictly increasing sequence $\left(n_k\right)$ in $\omega$, it follows that ${\{a_{n_k}-a_{n_{k+1}}\}}_{k\in \omega }$ is weakly p-summable and also a $(q^{\prime },q)$-limited set. Therefore, by p-$GPsc{P}_{(q^{\prime },q)}$, we have $a_{n_k}-a_{n_{k+1}}\to 0$ and hence S is Cauchy in E. Thus, A is a sequentially precompact subset of E.

(v)⇒(i) If $S={\left\{a_n\right\}}_{n\in \omega }$ is a weakly p-summable sequence in E which is a $(q^{\prime },q)$-limited set, then S is sequentially precompact in E. As S is also weakly null, we apply Lemma 1 to determine that S is a null sequence in E. □

Theorem 16 motivates the problem to characterize $(q^{\prime },q)$-limited p-convergent operators. Under some additional restrictions, this is achieved in the next assertion. If E and L are Banach spaces and $q^{\prime }=q=\infty$, the following theorem is proven in Theorem 2.1 of [30] and Theorem 1.1 of [45].

Theorem 17.

We let $2\le q^{\prime }\le q\le \infty$ and let E be a locally convex space with the Rosenthal property. Then, for operator T from E to a locally convex space L, the following assertions are equivalent:

(i)

T is $(q^{\prime },q)$-limited ∞-convergent;

(ii)

for each $(q^{\prime },q)$-limited set $A\subseteq E$, image $T\left(A\right)$ is sequentially precompact in L;

(iii)

for every locally convex (the same, normed) space H and each $(q^{\prime },q)$-limited operator $R:H\to E$, operator $T\circ R$ is sequentially precompact;

(iv)

for each $(q^{\prime },q)$-limited operator $R:{\ell }_1^0\to E$, operator $T\circ R$ is sequentially precompact.

If, in addition, E is locally complete, then (i)–(iv) are equivalent to the following:

(v)

for each $(q^{\prime },q)$-limited operator $R:{\ell }_1\to E$, operator $T\circ R$ is sequentially precompact.

Proof. 

(i)⇒(ii) We assume that $T\in \mathcal{L}(E,L)$ is $(q^{\prime },q)$-limited ∞-convergent and let A be a $(q^{\prime },q)$-limited subset of E. To show that $T\left(A\right)$ is sequentially precompact in L, we let ${\left\{a_n\right\}}_{n\in \omega }$ be a sequence in A. Since, by Theorem 4, A is weakly sequentially precompact, we can assume that ${\left\{a_n\right\}}_{n\in \omega }$ is weakly Cauchy. Therefore, for every strictly increasing sequence ${\left\{n_k\right\}}_{k\in \omega }$ in $\omega$, sequence ${\{a_{n_{k+1}}-a_{n_k}\}}_{k\in \omega }$ is weakly null and also a $(q^{\prime },q)$-limited set. Since T is $(q^{\prime },q)$-limited ∞-convergent, it follows that $T\left(a_{n_{k+1}}\right)-T\left(a_{n_k}\right)\to 0$ in L, and hence sequence ${\left\{T\left(a_{n_k}\right)\right\}}_{k\in \omega }$ is Cauchy in L. Thus $T\left(A\right)$ is sequentially precompact in E.

(ii)⇒(iii) We take $U\in {\mathcal{N}}_0\left(H\right)$ such that $R\left(U\right)$ is a $(q^{\prime },q)$-limited subset of E. By (ii), we know that $T\left(R\right(U\left)\right)$ is sequentially precompact in L. Thus, $T\circ R$ is a sequentially precompact operator.

(iii)⇒(iv) and (iii)⇒(v) are obvious.

(iv)⇒(i) and (v)⇒(i): We let $S={\left\{x_n\right\}}_{n\in \omega }$ be a weakly null sequence in E which is a $(q^{\prime },q)$-limited set. Since S is bounded, Proposition 14.9 of [17] implies that linear map $R:{\ell }_1^0\to E$ (or $R:{\ell }_1\to E$ if E is locally complete) defined by

$$R(a_0{e}_0+\cdots +a_n{e}_n):=a_0{x}_0+\cdots +a_n{x}_n(n\in \omega ,a_0,\dots ,a_n\in \mathbb{F}).$$

is continuous. It is clear that $R\left(B_{{\ell }_1^0}\right)\subseteq \overline{\mathrm{acx}}\left(S\right)$ (or $R\left(B_{{\ell }_1}\right)\subseteq \overline{\mathrm{acx}}\left(S\right)$ if E is locally complete). We observe that $\overline{\mathrm{acx}}\left(S\right)$ is also a $(q^{\prime },q)$-limited set. Therefore, operator R is $(q^{\prime },q)$-limited, and hence, by (iv) or (v), $T\circ R$ is sequentially precompact. In particular, the weakly null sequence $T\left(S\right)$ is (sequentially) precompact in L. Therefore, by Lemma 1, $T\left(x_n\right)\to 0$ in L. Thus, T is $(q^{\prime },q)$-limited ∞-convergent. □

Below, we obtain a sufficient condition on locally convex spaces to have p-$GPsc{P}_{(q,\infty )}$.

Proposition 15.

We let $p,q\in [1,\infty ]$ and let E be a locally convex space. Then, E has p-$GPsc{P}_{(q,\infty )}$ if one of the the following conditions holds:

(i)

for every locally convex space X and for each $T\in \mathcal{L}(X,E)$ whose adjoint $T^{*}:E_{w^{*}}^{\prime }\to X_{\beta }^{\prime }$ is q-convergent, operator T transforms the bounded subsets of X into sequentially precompact subsets of E;

(ii)

for every normed space X and for each $T\in \mathcal{L}(X,E)$ whose adjoint $T^{*}:E_{w^{*}}^{\prime }\to X_{\beta }^{\prime }$ is q-convergent, operator T is sequentially precompact;

(iii)

the same as (ii) with $X={\ell }_1^0$;

(iv)

if E is locally complete, the same as (ii) with X being a Banach space;

(v)

if E is locally complete, the same as (ii) with $X={\ell }_1$;

Proof. 

Since implications (i)⇒(ii)⇒(iii) and (ii)⇒(iv)⇒(v) are obvious, we prove that (iii) and (v) imply p-$GPsc{P}_{(q,\infty )}$. We suppose for a contradiction that E has no p-$GPsc{P}_{(q,\infty )}$. Then, there is a weakly p-summable sequence $S={\left\{x_n\right\}}_{n\in \omega }$ in E which is a $(q,\infty )$-limited set such that $x_n\nrightarrow 0$ in E. Without loss of generality, we assume that $x_n\notin U$ for some $U\in {\mathcal{N}}_0\left(E\right)$ and all $n\in \omega$.

Since S is a $(q,\infty )$-limited set, Proposition 5.9 of [27] implies that linear map $T:{\ell }_1^0\to E$ (or $T:{\ell }_1\to E$ if E is locally complete) defined by

$$T(a_0{e}_0+\cdots +a_n{e}_n):=a_0{x}_0+\cdots +a_n{x}_n(n\in \omega ,a_0,\dots ,a_n\in \mathbb{F}),$$

is continuous and its adjoint $T^{*}:E_{w^{*}}^{\prime }\to {\ell }_{\infty }$ is q-convergent. Therefore, by (iii) or (v), operator T is sequentially precompact. In particular, set $T\left({\left\{e_n\right\}}_{n\in \omega }\right)=S$ is (sequentially) precompact in E. Since S is also weakly null, Lemma 1 implies that $x_n\to 0$ in E. But this contradicts the choice of S. □

For an important case which covers all strict $\left(LF\right)$ spaces, we have the following characterization of $GPsc{P}_{(q,\infty )}$.

Theorem 18.

We let $2\le q\le \infty$. Then, for a locally convex space E with the Rosenthal property, the following assertions are equivalent:

(i)

E has $GPsc{P}_{(q,\infty )}$;

(ii)

for every locally convex space X and for each $T\in \mathcal{L}(X,E)$ whose adjoint $T^{*}:E_{w^{*}}^{\prime }\to X_{\beta }^{\prime }$ is q-convergent, operator T transforms the bounded subsets of X into sequentially precompact subsets of E;

(iii)

for every normed space X and for each $T\in \mathcal{L}(X,E)$ whose adjoint $T^{*}:E_{w^{*}}^{\prime }\to X_{\beta }^{\prime }$ is q-convergent, operator T is sequentially precompact;

(iv)

the same as (ii) with $X={\ell }_1^0$;

If, in addition, E is locally complete, then (i)–(iv) are equivalent to

(v)

the same as (iii) with X being a Banach space;

(vi)

the same as (iii) with $X={\ell }_1$.

Proof. 

By (the proof of) Proposition 15, we only have to prove implication (i)⇒(ii). So, we let X be a locally convex space and let $T\in \mathcal{L}(X,E)$ be such that the adjoint $T^{*}:E_{w^{*}}^{\prime }\to X_{\beta }^{\prime }$ is q-convergent. We have to show that for every bounded subset B of X, set $T\left(B\right)$ is sequentially precompact in E. To this end, we let ${\left\{{\chi }_n\right\}}_{n\in \omega }$ be a weak* q-summable sequence in $E^{\prime }$. As $T^{*}$ is q-convergent, we have $T^{*}\left({\chi }_n\right)\to 0$ in $X_{\beta }^{\prime }$. Since B is bounded, set $B^{\circ }$ is a neighborhood of zero in $X_{\beta }^{\prime }$. Therefore, for every $\epsilon >0$, there exists $N\in \omega$ such that $T^{*}\left({\chi }_n\right)\in \epsilon B^{\circ }$ for all $n\ge N$. Now, for every $y\in B$, we have

$$|\langle {\chi }_n,T\left(y\right)\rangle |=|\langle T^{*}\left({\chi }_n\right),y\rangle |\le \epsilon \mathrm{for}\mathrm{all}n\ge N.$$

Therefore, $T\left(B\right)$ is a $(q,\infty )$-limited set in E. Since $q\ge 2$ and E has the Rosenthal property, Theorem 4 implies that $T\left(B\right)$ is weakly sequentially precompact. To show that $T\left(B\right)$ is sequentially precompact in E, it suffices to prove that for any sequence $S={\left\{x_n\right\}}_{n\in \omega }$ in B, image $T\left(S\right)$ has a Cauchy subsequence. To this end, taking into account that $T\left(B\right)$ is weakly sequentially precompact, we can assume that $T\left(S\right)$ is weakly Cauchy. For any strictly increasing sequence ${\left\{n_k\right\}}_{k\in \omega }$ in $\omega$, sequence ${\{T\left(x_{n_k}\right)-T\left(x_{n_{k+1}}\right)\}}_{k\in \omega }$ is weakly null and a $(q,\infty )$-limited set. Therefore, by $GPsc{P}_{(q,\infty )}$ of E, we obtain $T\left(x_{n_k}\right)-T\left(x_{n_{k+1}}\right)\to 0$ in E. So ${\left\{T\left(x_{n_k}\right)\right\}}_{k\in \omega }$ is Cauchy in E. □

Now, we show that the class of spaces with p-$GPsc{P}_{(q^{\prime },q)}$ has nice stability properties.

Proposition 16.

We let $p,q,q^{\prime }\in [1,\infty ]$, $q^{\prime }\le q$ and let H be a subspace of a locally convex space E. If E has p-$GPsc{P}_{(q^{\prime },q)}$, then H also has p-$GPsc{P}_{(q^{\prime },q)}$.

Proof. 

We let $S={\left\{h_n\right\}}_{n\in \omega }$ be a weakly p-summable sequence in H which is also a $(q^{\prime },q)$-limited set. Then, by (vi) of Lemma 4.6 of [17], S is also weakly p-summable in E, and, by (iv) of Lemma 3.1 of [27], S is a $(q^{\prime },q)$-limited subset of E. By p-$GPsc{P}_{(q^{\prime },q)}$ of E, we have $h_n\to 0$ in E, and hence $h_n\to 0$ also in H. Thus, H has p-$GPsc{P}_{(q^{\prime },q)}$. □

Proposition 17.

We let $p,q,q^{\prime }\in [1,\infty ]$, $q^{\prime }\le q$ and let ${\left\{E_i\right\}}_{i\in I}$ be a mom-empty family of locally convex spaces. Then, the following assertions are equivalent:

(i)

space $E={\prod }_{i\in I}{E}_i$ has p-$GPsc{P}_{(q^{\prime },q)}$;

(ii)

space $L={⨁}_{i\in I}{E}_i$ has p-$GPsc{P}_{(q^{\prime },q)}$;

(iii)

for every $i\in I$, space $E_i$ has p-$GPsc{P}_{(q^{\prime },q)}$.

Proof. 

(i)⇒(ii) We let $S={\left\{x_n\right\}}_{n\in \omega }$ be a weakly p-summable sequence in L which is also a $(q^{\prime },q)$-limited set. Since S is bounded, there is a finite subset F of I such that $S\subseteq {\prod }_{i\in F}{E}_i\times {\prod }_{i\in I\setminus F}\left\{0_i\right\}$. Since ${\prod }_{i\in F}{E}_i$ is a subspace of E, Proposition 16 implies that $x_n\to 0$ in L. Thus, L has the p-$GPsc{P}_{(q^{\prime },q)}$.

(ii)⇒(iii) follows from Proposition 16 because any $E_i$ can be considered as a subspace of L.

(iii)⇒(i) We let $S={\{x_n=\left(x_{i,n}\right)\}}_{n\in \omega }$ be a weakly p-summable sequence in E which is also a $(q^{\prime },q)$-limited set. We observe that $x_n\to 0$ in E if and only if $x_{i,n}\to 0_i$ for every $i\in I$. For every $i\in I$, (ii) of Lemma 4.25 of [17] and (i) of Proposition 3.3 of [27] imply that sequence ${\left\{x_{i,n}\right\}}_{n\in \omega }$ is weakly p-summable in $E_i$, which is also a $(q^{\prime },q)$-limited set. Therefore, by the p-$GPsc{P}_{(q^{\prime },q)}$ of $E_i$, we obtain $x_{i,n}\to 0_i$ in $E_i$, as desired. □

6. Strong Versions of the ${\mathit{V}}^{*}$ Type Properties of Pełczyński

In Definition 13 of $V^{*}$ type sets, $(p,q)$-$\left(V^{*}\right)$ sets are relatively compact in the original topology instead of the weak topology. Therefore, we obtain “strong” versions of $V_{(p,q)}^{*}$ properties.

Definition 15.

We let $1\le p\le q\le \infty$. A locally convex space $(E,\tau )$ is said to have

• the strong $V_{(p,q)}^{*}$ property (resp., strong $E{V}_{(p,q)}^{*}$ property) if every $(p,q)$-$\left(V^{*}\right)$ (resp., $(p,q)$-$\left(E{V}^{*}\right)$) subset of E is relatively compact;
• the strong sequential $V_{(p,q)}^{*}$ property (resp., strong sequential $E{V}_{(p,q)}^{*}$ property) if every $(p,q)$-$\left(V^{*}\right)$ (resp., $(p,q)$-$\left(E{V}^{*}\right)$) subset of E is relatively sequentially compact; in short, the strong $s{V}_{(p,q)}^{*}$ property or the strong $sE{V}_{(p,q)}^{*}$ property;
• the strong precompact $V_{(p,q)}^{*}$ property (resp., strong precompact $E{V}_{(p,q)}^{*}$ property) if every $(p,q)$-$\left(V^{*}\right)$ (resp., $(p,q)$-$\left(E{V}^{*}\right)$) subset of E is τ-precompact; in short, the strong $pr{V}_{(p,q)}^{*}$ property or the strong $prE{V}_{(p,q)}^{*}$ property;
• the strong sequentially precompact $V_{(p,q)}^{*}$ property (resp., strong sequentially precompact $E{V}_{(p,q)}^{*}$ property) if every $(p,q)$-$\left(V^{*}\right)$ (resp., $(p,q)$-$\left(E{V}^{*}\right)$) subset of E is sequentially precompact in τ; in short, the strong $sp{V}_{(p,q)}^{*}$ property or the strong $spE{V}_{(p,q)}^{*}$ property.

If $q=\infty$, we say that E has the strong (resp., strong sequential, precompact, or sequentially precompact) $V_p^{*}$ property. If $p=1$ and $q=\infty$, then we say that E has the strong (resp., strong sequential, precompact, or sequentially precompact) $V^{*}$ property.

The next lemma follows from the corresponding definitions and the simple fact that every $(p,q)$-limited set is also a $(p,q)$-$\left(V^{*}\right)$ set.

Lemma 7.

We let $1\le p\le q\le \infty$ and let E be a locally convex space. Then,

(i)

if E has the strong $V_{(p,q)}^{*}$ property, then it has the $G{P}_{(p,q)}$ property;

(ii)

if E has the strong $s{V}_{(p,q)}^{*}$ property, then it has the $sG{P}_{(p,q)}$ property;

(iii)

if E has the strong $pr{V}_{(p,q)}^{*}$ property, then it has the $prG{P}_{(p,q)}$ property;

(iv)

if $E=E_w$, then E has the strong $pr{V}_{(p,q)}^{*}$ property.

Remark 7.

The converse in Lemma 7 is not true in general even for Banach spaces. Indeed, we let $p=q=\infty$. Then, by Theorem 2, $c_0$ has the Gelfand–Phillips property. However, $c_0$ does not have the strong $V_{\infty }^{*}$ property because $B_{c_0}$ is not compact but it is an ∞-$\left(V^{*}\right)$ set since any weakly null sequence in ${\ell }_1={\left(c_0\right)}_{\beta }^{\prime }$, by the Schur property, is norm null.

The next proposition shows that for a wide class of locally convex spaces, including all strict $\left(LF\right)$-spaces, all strong $V^{*}$ type properties coincide.

Proposition 18.

We let $1\le p\le q\le \infty$ and let E be a von Neumann complete, p-quasibarrelled angelic space (for example, E is a strict $\left(LF\right)$-space). Then, all the properties of strong $V_{(p,q)}^{*}$, strong $E{V}_{(p,q)}^{*}$, strong $s{V}_{(p,q)}^{*}$, strong $sE{V}_{(p,q)}^{*}$, strong $pr{V}_{(p,q)}^{*}$, strong $prE{V}_{(p,q)}^{*}$, strong $sp{V}_{(p,q)}^{*}$, or strong $spE{V}_{(p,q)}^{*}$ are equivalent.

Proof. 

Since E is p-quasibarrelled, Lemma 7.2 of [17] implies that the family of $(p,q)$-$\left(V^{*}\right)$ sets coincides with the family of $(p,q)$-$\left(E{V}^{*}\right)$ sets. Therefore, all equicontinuous $V^{*}$ type properties coincide with the corresponding $V^{*}$ type property. Since E is von Neumann complete, it is clear that the strong $pr{V}_{(p,q)}^{*}$ property coincides with the strong $V_{(p,q)}^{*}$ property. Finally, the angelicity of E implies that the (relatively) compact subsets of E are exactly the (relatively) sequentially compact sets. Taking into account that the closure of a $(p,q)$-$\left(V^{*}\right)$ set is a $(p,q)$-$\left(V^{*}\right)$ set, it follows that the strong $V_{(p,q)}^{*}$ (resp., $pr{V}_{(p,q)}^{*}$) property coincides with the strong $s{V}_{(p,q)}^{*}$ (resp., strong $sp{V}_{(p,q)}^{*}$) property. □

The following two propositions show that the classes of locally convex spaces with strong $V_{(p,q)}^{*}$ type properties are stable under taking direct products, direct sums and closed subspaces. We omit their proofs because they can be obtained from the proofs of Propositions 8 and 9, respectively, just replacing “$(p,q)$-limited” by “$(p,q)$-$\left(V^{*}\right)$” and from Proposition 7.4 of [17] which describes $(p,q)$-$\left(V^{*}\right)$ sets in direct products and direct sums.

Proposition 19.

We let $1\le p\le q\le \infty$ and let ${\left\{E_i\right\}}_{i\in I}$ be a non-empty family of locally convex spaces. Then,

(i)

$E={\prod }_{i\in I}{E}_i$ has the strong $V_{(p,q)}^{*}$ (resp., strong $E{V}_{(p,q)}^{*}$, strong $pr{V}_{(p,q)}^{*}$, or strong $prE{V}_{(p,q)}^{*}$) property if and only if all spaces $E_i$ have the same property;

(ii)

$E={⨁}_{i\in I}{E}_i$ has the strong $V_{(p,q)}^{*}$ (resp., strong $E{V}_{(p,q)}^{*}$, strong $s{V}_{(p,q)}^{*}$, strong $sE{V}_{(p,q)}^{*}$, strong $pr{V}_{(p,q)}^{*}$, strong $prE{V}_{(p,q)}^{*}$, strong $sp{V}_{(p,q)}^{*}$, or strong $spE{V}_{(p,q)}^{*}$) property if and only if all spaces $E_i$ have the same property;

(iii)

if $I=\omega$ is countable, then $E={\prod }_{i\in \omega }{E}_i$ has the strong $s{V}_{(p,q)}^{*}$ (resp., strong $sE{V}_{(p,q)}^{*}$, strong $sp{V}_{(p,q)}^{*}$, or strong $spE{V}_{(p,q)}^{*}$) property if and only if all spaces $E_i$ have the same property.

Proposition 20.

We let $1\le p\le q\le \infty$ and let L be a subspace of a locally convex space E.

(i)

If L is closed in E and E has the the strong $V_{(p,q)}^{*}$ (resp., strong $E{V}_{(p,q)}^{*}$, strong $pr{V}_{(p,q)}^{*}$, or strong $prE{V}_{(p,q)}^{*}$) property, then L also has the same property.

(ii)

If L is sequentially closed in E and E has the strong $s{V}_{(p,q)}^{*}$ (resp., strong $sE{V}_{(p,q)}^{*}$) property, then L also has the same property.

(iii)

If E has the strong $pr{V}_{(p,q)}^{*}$ (resp., strong $prE{V}_{(p,q)}^{*}$, strong $sp{V}_{(p,q)}^{*}$, or strong $spE{V}_{(p,q)}^{*}$) property, then L also has the same property.

Below, we characterize locally convex spaces with the strong precompact $V_p^{*}$ property and the strong sequentially precompact $V_p^{*}$ property.

Theorem 19.

We let $p\in [1,\infty ]$ and let E be a locally convex space. Then, the following assertions are equivalent:

(i)

E has the strong (resp., sequentially) precompact $V_p^{*}$ property;

(ii)

for every locally convex space X, each $T\in \mathcal{L}(X,E)$ whose adjoint $T^{*}:E_{\beta }^{\prime }\to X_{\beta }^{\prime }$ is p-convergent, transforms bounded sets of X into (resp., sequentially) precompact subsets of E;

(iii)

for every normed space X and each $T\in \mathcal{L}(X,E)$, if the adjoint $T^{*}:E_{\beta }^{\prime }\to X_{\beta }^{\prime }$ is p-convergent, then T is (resp., sequentially) precompact;

(iv)

the same as (iii) with $X={\ell }_1^0$.

If, in addition, E is locally complete, then (i)–(iv) are equivalent to

(v)

the same as (iii) with X a Banach space;

(vi)

the same as (iii) with $X={\ell }_1$.

Proof. 

(i)⇒(ii) We let X be a locally convex space and let $T\in \mathcal{L}(X,E)$ be such that the adjoint $T^{*}:E_{\beta }^{\prime }\to X_{\beta }^{\prime }$ is p-convergent. We have to show that set $T\left(B\right)$ is (sequentially) precompact in E for every bounded subset B of X. To this end, we let ${\left\{{\chi }_n\right\}}_{n\in \omega }$ be a weakly p-summable sequence in $E_{\beta }^{\prime }$. As $T^{*}$ is p-convergent, we have $T^{*}\left({\chi }_n\right)\to 0$ in $X_{\beta }^{\prime }$. Since B is bounded, set $B^{\circ }$ is a neighborhood of zero in $X_{\beta }^{\prime }$. Therefore, for every $\epsilon >0$, there exists $N\in \omega$ such that $T^{*}\left({\chi }_n\right)\in \epsilon B^{\circ }$ for all $n\ge N$. Now, for every $y\in B$, we have

$$|\langle {\chi }_n,T\left(y\right)\rangle |=|\langle T^{*}\left({\chi }_n\right),y\rangle |\le \epsilon \mathrm{for}\mathrm{all}n\ge N.$$

Therefore, $T\left(B\right)$ is a p-$\left(V^{*}\right)$ set in E. From the strong (sequentially) precompact $V_p^{*}$ property, it follows that $T\left(B\right)$ is (sequentially) precompact in E, as desired.

(ii)⇒(iii)⇒(iv) and (iii)⇒(v)⇒(vi) are obvious.

(iv)⇒(i) and (vi)⇒(i): To show that E has the strong (sequentially) precompact $V_p^{*}$ property, we assume for a contradiction that there is a p-$\left(V^{*}\right)$ set A in E which is not (resp., sequentially) precompact. Then, there are $U\in {\mathcal{N}}_0\left(E\right)$ and a U-separated sequence $S={\left\{x_n\right\}}_{n\in \omega }$ in A (resp., a sequence $S={\left\{x_n\right\}}_{n\in \omega }$ in A which does not have a Cauchy subsequence). Since S is also a p-$\left(V^{*}\right)$ set, Proposition 14.9 of [17] implies that linear map $T:{\ell }_1^0\to E$ (or $T:{\ell }_1\to E$ if E is locally complete) defined by

$$T(a_0{e}_0+\cdots +a_n{e}_n):=a_0{x}_0+\cdots +a_n{x}_n(n\in \omega ,a_0,\dots ,a_n\in \mathbb{F}),$$

is continuous and its adjoint $T^{*}:E_{\beta }^{\prime }\to {\ell }_{\infty }$ is p-convergent. Therefore, by (iv) or (vi), operator T is (resp., sequentially) precompact. In particular, set $T\left({\left\{e_n\right\}}_{n\in \omega }\right)=S$ is (sequentially) precompact in E, which contradicts the choice of S. □