In this paper we introduce the notion fuzzy soft metric space after that we define open fuzzy soft ball, fuzzy soft bounded set, fuzzy soft convergence of sequences, fuzzy soft continuous function from a fuzzy soft metric space to another fuzzy metric space then we study this space and some basic results are investigated.

Our goal in the present paper is to introduce a new type of fuzzy inner product space. After that, to illustrate this notion, some examples are introduced. Then we prove that that every fuzzy inner product space is a fuzzy normed space. We also prove that the cross product of two fuzzy inner spaces is again a fuzzy inner product space. Next, we prove that the fuzzy inner product is a non decreasing function. Finally, if U is a fuzzy complete fuzzy inner product space and D is a fuzzy closed subspace of U, then we prove that U can be written as a direct sum of D and the fuzzy orthogonal complement    of D.

In this paper we recall the definition of fuzzy norm then basic properties of fuzzy  normed space is recalled after that we introduced the definition of compact fuzzy  normed space. Then basic properties of compact fuzzy normed space is proved.

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The definition of fuzzy length space on fuzzy set in this research was introduced after the studies and discussion of many properties of this space were proved, and then an example to illustrate this notion was given. Also the definition of fuzzy convergence, fuzzy bounded fuzzy set, and fuzzy dense fuzzy set space was introduced, and then the definition of fuzzy continuous operator was introduced.