Absence of Non-Trivial Fuzzy Inner Product Spaces and the Cauchy–Schwartz Inequality

Abstract

First, we show that the non-trivial fuzzy inner product space under the linearity condition does not exist, which means a fuzzy inner product space with linearity produces only a crisp real number for each pair of vectors. If the positive-definiteness is added to the condition, then the Cauchy–Schwartz inequality is also proved.

Fuzzy real weak inner product: A new approach

“…In [18], it is mentioned that the author couldn't prove the converse of their proposed theorem would hold in terms of fuzzy context except the crisp case (see the subsection 3.1). In our previous study [3], we have proved that the converse in terms of fuzzy context can not hold. In fact, it was shown that a fuzzy inner product on a vector space, producing a fuzzy real number as its value for each tuple of elements in the space, is merely the embedding of a real valued inner product into the fuzzy number systems over the set of tuples consisting of elements in the space because of its linear property.…”

“…But, this fuzzy real inner product is a trivial one in the sense that it is the embedding of a real-valued inner product into the fuzzy real number systems over the set of tuples consisting of elements of a give vector space, which is shown in [3] or see the subsection 3.2. Thus we need to change the conditions of Definition 2.6 into weak ones.…”

“…What is the concept of fuzzy inner products? Some authors makes various module concepts where their proposed fuzzy inner product takes its values, while we consider a fuzzy inner product which takes its values in F (R) (see [3]). The comparison is given as follows:…”

“…Saheli [15]- [17] conducted a comparative study of the relationship between Goudarzias definition and Hasankhanis definition for a fuzzy inner product. Byun et al [3] showed the absence of non-trivial fuzzy inner product spaces and showed Cauchy-Schwartz inequality. Yoon et al [7] proposed a new idea of classification of complex fuzzy numbers and dealt with fuzzy inner products.…”

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There are various kinds of fuzzy inner products. It was shown [3] that a fuzzy inner product on a vector space, producing a fuzzy real number as its value, is merely the embedding of a real valued inner product into the fuzzy number systems over the set of tuples consisting of elements in the space because of its linear property. Thus the linear property is needed to be weakened.

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