Abstract
In this paper the definition of fuzzy normed space is recalled and its basic properties. Then the definition of fuzzy compact operator from fuzzy normed space into another fuzzy normed space is introduced after that the proof of an operator is fuzzy compact if and only if the image of any fuzzy bounded sequence contains a convergent subsequence is given. At this point the basic properties of the vector space FC(V,U)of all fuzzy compact linear operators are investigated such as when U is complete and the sequence ( ) of fuzzy compact operators converges to an operator T then T must be fuzzy compact. Furthermore we see that when T is a fuzzy compact operator and S is a fuzzy bounded operator then the composition TS and ST are fuzzy compact operators. Finally, if T belongs to FC(V,U) and dimension of V is finite then T is fuzzy compact is proved.
Highlights
Some results of fuzzy complete fuzzy normed spaces were studied by Saadati and Vaezpour in 2005 (1)
Properties of fuzzy bounded linear operators on a fuzzy normed space were investigated by Bag and Samanta in 2005 (2).The fuzzy normed linear space and its fuzzy topological structure were studied by Sadeqi and Kia in 2009 (3)
Properties of fuzzy continuous operators on a fuzzy normed linear spaces were studied by Nadaban in 2015 (4)
Summary
Some results of fuzzy complete fuzzy normed spaces were studied by Saadati and Vaezpour in 2005 (1). Definition 8 :(5) A sequence(an) in a fuzzy normed space(V, LV,∗) is called converges to a ∈ V if for each q > 0 and t > 0 we can find N∈ N with LV[an − a, t] > (1 − q) for all n ≥ N. Suppose that (V, LV, ⨂) and(W, LW, ⨀) are two fuzzy normed spaces .The operator S: V → W is said to be fuzzy continuous at v0 ∈ V if for all t > 0 and for all 0 < α < 1there is s and there is β with, LV[v − v0, s] > (1 − β) we have LW[S(v) − S(v0), t] > (1 − α) for all v ∈V.
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