Fuzzy Inner Product Space: Literature Review and a New Approach

Lorena Popa,Lavinia Sida

doi:10.3390/math9070765

Abstract

The aim of this paper is to provide a suitable definition for the concept of fuzzy inner product space. In order to achieve this, we firstly focused on various approaches from the already-existent literature. Due to the emergence of various studies on fuzzy inner product spaces, it is necessary to make a comprehensive overview of the published papers on the aforementioned subject in order to facilitate subsequent research. Then we considered another approach to the notion of fuzzy inner product starting from P. Majundar and S.K. Samanta’s definition. In fact, we changed their definition and we proved some new properties of the fuzzy inner product function. We also proved that this fuzzy inner product generates a fuzzy norm of the type Nădăban-Dzitac. Finally, some challenges are given.

Highlights

[15] A fuzzy inner product space (FIP-space) is a triplet (X; P; ∗), where X is a real linear space, ∗ is a continuous t-norm and P is a fuzzy set on X2 × R satisfying the following conditions for every x; y; z ∈ X and t ∈ R
[16] A fuzzy inner product space (FIP - space) is a triplet (X, P, ∗), where X is a real linear space, ∗ is a continuous t-norm and P is a fuzzy set in X × X × R s.t. the following conditions hold for every x, y, z ∈ X and s, t, r ∈ R
We wrote a literature review regarding the diverse approaches of fuzzy inner product space concept, but we introduced a new approach

Summary

Introduction

Goudarzi and S.M. Vaezpour [16] alter the definition of the fuzzy inner product space and prove several interesting results which take place in each fuzzy inner product space. The disadvantage of this definition is that only linear spaces over R can be considered. This paper is organized as follows—in Section 2 we make a literature review Such an approach is deemed useful for the readers as it would enable them to better understand the evolution of the fuzzy inner product space concepts. Majumdar and S.K. Samanta’s definition of inner product space and we introduced and proved some new properties of the fuzzy inner product function.

Preliminaries

Conclusions and Future Works