Fuzzy real weak inner product: A new approach

Abstract There are various kinds of fuzzy inner products. It was shown \cite{BLLY} 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. In this paper, a new concept, called a \emph{fuzzy real weak inner product} on a vector space, is introduced and some nontrivial examples are given. Then, a fuzzy real weak inner product in a suitable condition is deeply studied, which may be either almost positive-definite or positive-definite. Furthermore, it satisfies both Cauchy inequality and Parallelogram law even though it is not linear. And some properties of a general type of a fuzzy real weak inner product, especially the approximation related to the Parallelogram law, are investigated in analytic viewpoint.