Some Properties of Fuzzy Inner Product Space

Abstract

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.

Properties of The Space GFB(V, U)

“…is a nondecreasing function of t in the general fuzzy normed space (V, , ,⊗) for all u V this means when 0< t <s implies (u, t) < (u, s) . Remark 2.13: [28] Assume the general fuzzy normed space (V, , ,⊗). Then for any u V, s>0 ,0<n<1 .…”

“…A subset W V is called a general fuzzy closed set if is a general fuzzy open. Definition 2.16: [28] Suppose that (V, , ,⊗) is a general fuzzy normed space. A sequence ( ) in V is said to be general fuzzy approaches to u if every 0 < < 1 and 0< s there is N such that ( u , s) > (1 ) for every n≥ N. If ( ) is general fuzzy approaches to the fuzzy limit u we write m = u or u.…”

Some Properties of Fuzzy Soft Metric Space

“…Kider and Kadhum in 2017 [11] introduce the fuzzy norm for a fuzzy bounded operator on a fuzzy normed space and proved its basic properties then other properties was proved by Kadhum in 2017 [12]. Ali in 2018 [13] proved basic properties of complete fuzzy normed algebra.…”

“…Ali in 2018 [13] proved basic properties of complete fuzzy normed algebra. Kider and Ali in 2018 [14] introduce the notion of fuzzy absolute value and study properties of finite dimensional fuzzy normed space.…”

“…The concept of general fuzzy normed space were presented by Kider and Gheeab in 2019 [15] [16] also they proved basic properties of this space and the general fuzzy normed space GFB(V, U). Kider and Kadhum in 2019 [17] introduce the notion fuzzy compact linear operator and proved its basic properties.…”