Embedding the Different Families of Fuzzy Sets into Banach Spaces by Using Cauchy Sequences

The family $\mathcal{F}\left(U\right)$ of all fuzzy sets in a normed space cannot be a vector space. This deficiency affects its application to practical problems. The topics of functional analysis and nonlinear analysis in mathematics are based on vector spaces. Since these two topics have been well developed such that their tools can be used to solve practical economics and engineering problems, lacking a vector structure for the family $\mathcal{F}\left(U\right)$ diminishes its applications to these kinds of practical problems when fuzzy uncertainty has been detected in a real environment. Embedding the whole family $\mathcal{F}\left(U\right)$ into a Banach space is still not possible. However, it is possible to embed some interesting and important subfamilies of $\mathcal{F}\left(U\right)$ into some suitable Banach spaces. This paper presents the concrete and detailed structures of these kinds of Banach spaces such that their mathematical structures can penetrate the core of practical economics and engineering problems in fuzzy environments. The important issue of uniqueness for these Banach spaces is also addressed via the concept of isometry.