**1. Introduction**

The research papers of A.K. Katsaras [1,2] laid the foundations of the fuzzy functional analysis. Moreover, he was the first one who introduced the concept of a fuzzy norm. This concept has amassed grea<sup>t</sup> interest among mathematicians. Thus, in 1992, C. Felbin [3] introduced a new idea of a fuzzy norm in a linear space by associating a real fuzzy number to each element of the linear space. In 2003, T. Bag and S.K. Samanta [4] put forward a new concept of a fuzzy norm, which was a fuzzy set on *X* × R. New fuzzy norm concepts were later introduced by R. Saadati and S.M. Vaezpour [5], C. Alegre and S.T. Romaguera [6], R. Ameri [7], I. Gole¸t [8], A.K. Mirmostafaee [9]. In this paper we use the definition introduced by S. N ˘ad ˘aban and I. Dzitac [10].

Hilbert spaces lay at the core of functional analysis. They frequently and naturally appear in the fields of Mathematics and Physics, being indispensable in the theory of differential equations, quantum mechanics, quantum logic, quantum computing, the Fourier analysis (with applications in the signal theory). Therefore, many mathematicians have focused on finding an adequate definition of the fuzzy inner product space. Although there are many research papers focused on the concept of a fuzzy norm and its diverse applications, there are few papers which study the concept of a fuzzy inner product.

Even though there are few results, we consider that every breakthrough has been an important one. Moreover, we are certain that there exists a correct definition of the fuzzy inner product that when discovered, will not only generate a worldwide consensus on this subject but trigger countless applications in various fields.

The importance of this subject has led us to write this paper and we have tried to further the knowledge on the matter.

Furthermore, we wish to mention the main results of the already existent literature.

R. Biswas in [11] defined the fuzzy inner product of elements in a linear space and two years later J.K. Kohli and R. Kumar altered the Biswas's definition of inner product space [12]. In fact, they showed that the definition of a fuzzy inner product space in terms of the conjugate of a vector is redundant and that those definitions are only restricted to the real linear spaces. They also introduced the fuzzy co-inner product spaces and the fuzzy co-norm functions in their paper.

Two years later, in 1995, Eui-Whan Cho, Young-Key Kim and Chae-Seob Shin introduced and defined in [13] a fuzzy semi-inner-product space and investigated some properties of this fuzzy semi inner product space, those definitions are not restricted to the real linear spaces.

In 2008, P. Majumdar and S.K. Samanta [14] succeeded in taking the first step towards finding a reliable definition of a fuzzy inner product space. From their definition we can identify a serious problem in regard to finding a new reliable definition for the fuzzy inner product. The classical inequality Cauchy-Schwartz cannot be obtained by applying the other axioms and thus had to be introduced itself as an axiom (axiom (FIP2)).

In 2009, M. Goudarzi, S.M. Vaezpour and R. Saadati [15] introduced the concept of intuitionistic fuzzy inner product space. In this context, the Cauchy-Schwartz inequality, the Pythagorean Theorem and some convergence theorems were established.

In the same year, M. 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. More specifically, they introduced the notion of a fuzzy Hilbert space and deduce a fuzzy version of Riesz representation theorem.

In 2013, S. Mukherjee and T.Bag [17] amends the definition put forward by M.Goudarzi and S.M. Vaezpour by discarding the (FI-6) condition and by enacting minor changes to the (FI-4) and (FI-5) conditions.

In 2010, A. Hasankhani, A. Nazari and M. Saheli [18] introduced a new concept of a fuzzy Hilbert space. This concept is entirely different from the previous ones as this fuzzy inner product generates a new fuzzy norm of type Felbin.

The disadvantage of this definition is that only linear spaces over R can be considered. Another disadvantage is the difficulty of working with real fuzzy numbers.

In the subsequent years, many papers addressing this theme were published (see [7,19–24]).

In 2016, M. Saheli and S.Khajepour Gelousalar [25] modified the definition of the fuzzy inner product space and proved some properties of the new fuzzy inner product space.

Also, in 2016, Z. Solimani and B. Daraby [26] slightly altered the definition of a fuzzy scalar product introduced in [18] by changing the (IP2) condition and merging the (IP4) and (IP5).

In 2017, E. Mostofian, M. Azhini and A. Bodaghi [27] presented two new concepts of fuzzy inner product spaces and investigated some of basic properties of these spaces.

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. Thus, this section can constitute a starting point for other mathematicians interested in this subject. In Section 3 we introduce a new definition of the fuzzy inner product space starting from P. Majumdar and S.K. Samanta's definition [14]. In fact, we modified the P. 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. This paper ends up with some conclusions and future works in Section 4.
