**2. Preliminaries**

**Definition 1.** *[10] Let X be a vector space over a field* K *and* ∗ *be a continuous t-norm. A fuzzy set N in X* × [0, ∞] *is called a fuzzy norm on X if it satisfies:*


*(N5)* (∀)*x* ∈ *X*, *<sup>N</sup>*(*<sup>x</sup>*, ·) *is left continuous and* lim*t*→∞*<sup>N</sup>*(*<sup>x</sup>*, *t*) = 1*.*

*The triplet* (*<sup>X</sup>*, *N*, ∗) *will be called fuzzy normed linear space (briefly FNLS).*

**Definition 2.** *[14] A fuzzy inner product space (FIP-space) is a pair* (*<sup>X</sup>*, *<sup>P</sup>*)*, where X is a linear space over* C *and P is a fuzzy set in X* × *X* × C *s.t.*

*(FIP1) For s*, *t* ∈ C, *<sup>P</sup>*(*x* + *y*, *z*, |*t*| + |*s*|) ≥ min{*P*(*<sup>x</sup>*, *z*, |*t*|), *<sup>P</sup>*(*y*, *z*, |*s*|)}*; (FIP2) For s*, *t* ∈ C, *<sup>P</sup>*(*<sup>x</sup>*, *y*, |*st*|) ≥ min*P<sup>x</sup>*, *x*, |*s*|<sup>2</sup>, *<sup>P</sup><sup>y</sup>*, *y*, |*t*|<sup>2</sup>*; (FIP3) For t* ∈ C, *<sup>P</sup>*(*<sup>x</sup>*, *y*, *t*) = *<sup>P</sup><sup>y</sup>*, *x*, *t; (FIP4) <sup>P</sup>*(*αx*, *y*, *t*) = *Px*, *y*, *t*|*α*|, *t* ∈ C, *α* ∈ C<sup>∗</sup>*; (FIP5) <sup>P</sup>*(*<sup>x</sup>*, *x*, *t*) = 0,(∀)*t* ∈ C \ R+*; (FIP6)* [*P*(*<sup>x</sup>*, *x*, *t*) = 1,(∀)*t* > 0] *iff x* = 0*; (FIP7) <sup>P</sup>*(*<sup>x</sup>*, *x*, ·) : R → [0, 1] *is a monotonic non-decreasing function of* R *and* lim*t*→∞ *<sup>P</sup>*(*<sup>x</sup>*, *x*, *t*) = 1*.*

*P will be called the fuzzy inner product on X.*

**Definition 3.** *[15] A fuzzy inner product space (FIP-space) is a triplet* (*<sup>X</sup>*; *P*; <sup>∗</sup>)*, where X is a real linear space,* ∗ *is a continuous t-norm and P is a fuzzy set on X*<sup>2</sup> × R *satisfying the following conditions for every x*; *y*; *z* ∈ *X and t* ∈ R.

> *if t* > 0

*if t* ≤ 0*;*

*i f α* > 0

*i f α* = 0

*i f α* < 0*;*

*(FIP1) <sup>P</sup>*(*<sup>x</sup>*, *y*, 0) = 0*; (FIP2) <sup>P</sup>*(*<sup>x</sup>*, *y*, *t*) = *<sup>P</sup>*(*y*, *x*, *t*)*; (FIP3) <sup>P</sup>*(*<sup>x</sup>*, *x*, *t*) = *<sup>H</sup>*(*t*), ∀*t* ∈ R *iff x* = 0*, where H*(*t*) = 1, 0, *(FIP4) For any real number α*, *<sup>P</sup>*(*αx*, *y*, *t*) = ⎧⎪⎨⎪⎩*Px*, *y*, *tα* , *<sup>H</sup>*(*t*), 1 − *<sup>P</sup><sup>x</sup>*, *y*, *t* −*α* , *(FIP5)*

 sup *<sup>s</sup>*+*r*=*t P*(*<sup>x</sup>*, *<sup>z</sup>*,*<sup>s</sup>*) ∗ *<sup>P</sup>*(*y*, *<sup>z</sup>*,*<sup>r</sup>*)= *<sup>P</sup>*(*x* + *y*, *z*, *t*)*; (FIP6) <sup>P</sup>*(*<sup>x</sup>*, *y*, ·) : R → [0, 1] *is continuous on* R \ {0}*; (FIP7)* lim *t*→∞*<sup>P</sup>*(*<sup>x</sup>*, *y*, *t*) = 1*.*

**Definition 4.** *[16] A fuzzy inner product space (FIP - space) is a triplet* (*<sup>X</sup>*, *P*, <sup>∗</sup>)*, 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*,*<sup>r</sup>* ∈ R*.*

*(FI-1) <sup>P</sup>*(*<sup>x</sup>*, *x*, 0) = 0 *and <sup>P</sup>*(*<sup>x</sup>*, *x*, *t*) > 0, (∀)*t* > 0*; (FI-2) <sup>P</sup>*(*<sup>x</sup>*, *x*, *t*) = *H*(*t*) *for same t* ∈ R *iff x* = 0*; (FI-3) <sup>P</sup>*(*<sup>x</sup>*, *y*, *t*) = *<sup>P</sup>*(*y*, *x*, *t*)*; (FI-4) For any real number α*, *<sup>P</sup>*(*αx*, *y*, *t*) = ⎧⎪⎨⎪⎩*Px*, *y*, *tα* , *i f α* > 0 *<sup>H</sup>*(*t*), *i f α* = 0 1 − *<sup>P</sup><sup>x</sup>*, *y*, *t* −*α* , *i f α* < 0*; (FI-5)* sup *<sup>s</sup>*+*r*=*t P*(*<sup>x</sup>*, *<sup>z</sup>*,*<sup>s</sup>*) ∗ *<sup>P</sup>*(*y*, *<sup>z</sup>*,*<sup>r</sup>*) = *<sup>P</sup>*(*x* + *y*, *z*, *t*)*; (FI-6) <sup>P</sup>*(*<sup>x</sup>*, *y*, ·) : R → [0, 1] *is continuous on* R \ {0}*; (FI-7)* lim *t*→∞ *<sup>P</sup>*(*<sup>x</sup>*, *y*, *t*) = 1*.*

**Definition 5.** *[14] Let X be a linear space over* R*. A fuzzy set P in X* × *X* × R *is called fuzzy real inner product on X if* (∀)*<sup>x</sup>*, *y*, *z* ∈ *X and t* ∈ R*, the following conditions hold:*

*(FI-1) <sup>P</sup>*(*<sup>x</sup>*, *x*, 0) = 0, (∀)*t* < 0*; (FI-2) <sup>P</sup>*(*<sup>x</sup>*, *x*, *t*) = 1, (∀)*t* > 0 *iff x* = 0*; (FI-3) <sup>P</sup>*(*<sup>x</sup>*, *y*, *t*) = *<sup>P</sup>*(*y*, *x*, *t*)*;*

$$\begin{aligned} (FI\text{-}4) \ P(\text{ax}, y, t) &= \begin{cases} P\left(\text{x}, y, \frac{t}{\alpha}\right), & \text{if } \alpha > 0\\ H(t), & \text{if } \alpha = 0;\\ 1 - P\left(\text{x}, y, \frac{t}{\alpha}\right), & \text{if } \alpha < 0 \end{cases} \\\ (FI\text{-}5) \ P(\text{x} + y, z, t + s) &\geq \min\{P(\text{x}, z, t), P(y, z, s)\};\end{aligned}$$

$$(FI\text{-}6)\lim\_{t\to\infty}P(x\_\prime y\_\prime t) = 1.$$

*The pair* (*<sup>X</sup>*, *P*) *is called fuzzy real inner space.*

In order to present the definition of A. Hasankhani, A. Nazari and M. Saheli, we firstly need to define some concepts.

**Definition 6.** *[28] A fuzzy set in* R*, namely a mapping x* : R → [0, 1]*, with the following properties:*


*is called fuzzy real number. We denote by* F(R) *the set of all fuzzy real numbers.*

**Definition 7.** *[29] The arithmetic operation* +, −, ·, / *on* F(R) *are defined by:*

$$\begin{array}{l} \left(\mathbf{x} + \mathbf{y}\right)(t) = \bigvee\_{s \in \mathbb{R}} \min\left\{\mathbf{x}(s), y(t - s)\right\}, \ (\forall) t \in \mathbb{R};\\ \left(\mathbf{x} - \mathbf{y}\right)(t) = \bigvee\_{s \in \mathbb{R}} \min\left\{\mathbf{x}(s), y(s - t)\right\}, \ (\forall) t \in \mathbb{R};\\ \left(\mathbf{x}y\right)(t) = \bigvee\_{s \in \mathbb{R}^\*} \min\left\{\mathbf{x}(s), y(t/s)\right\}, \ (\forall) t \in \mathbb{R};\\ \left(\mathbf{x}/y\right)(t) = \bigvee\_{s \in \mathbb{R}^\*} \min\left\{\mathbf{x}(ts), y(s)\right\}, \ (\forall) t \in \mathbb{R}. \end{array}$$

**Remark 1.** *Let x* ∈ F(R) *and α* ∈ (0, 1]. *The α-level sets* [*x*]*α* = *t* ∈ R : *x*(*t*) ≥ *α are closed intervals <sup>x</sup>*<sup>−</sup>*α* , *<sup>x</sup>*+*α .*

**Definition 8.** *[18] Let X be a linear space over* R*. A fuzzy inner product on X is a mapping* < ·, · >: *X* × *X* → F(R) *s.t.*(∀)*<sup>x</sup>*, *y*, *z* ∈ *<sup>X</sup>*,(∀)*<sup>r</sup>* ∈ R*, we have:*

*(IP1)* < *x* + *y*, *z* >=< *x*, *z* > ⊕ < *y*, *z* >*; (IP2)* < *rx*, *y* >= *r*˜ < *x*, *y* >*, where r*˜ = 1, *if t* = *r* 0, *if t* = *r; (IP3)* < *x*, *y* >=< *y*, *x* >*; (IP4)* < *x*, *x* >≥ 0*; (IP5)* inf *α*<sup>∈</sup>(0,1] < *x*, *x* ><sup>−</sup>*α* = 0 *if x* = 0*; (IP6)* < *x*, *x* >= 0 ˜ *iff x* = 0*. The pair* (*<sup>X</sup>*,·, · ) *is called fuzzy inner product space.*

**Definition 9.** *[25] A fuzzy inner product space is a triplet* (*<sup>X</sup>*, *P*, <sup>∗</sup>)*, where X is a fuzzy set in X* × *X* × R *satisfying the following conditions for every x*, *y*, *z* ∈ *X and t*,*<sup>s</sup>* ∈ R*:*

*(FI1) <sup>P</sup>*(*<sup>x</sup>*, *y*, 0) = 0*; (FI2) <sup>P</sup>*(*<sup>x</sup>*, *y*, *t*) = *<sup>P</sup>*(*y*, *x*, *t*)*; (FI3) <sup>P</sup>*(*<sup>x</sup>*, *x*, *t*) = 1, (∀)*t* > 0 *iff x* = 0*; (FI4)* (∀)*α* ∈ R, *t* = 0, *<sup>P</sup>*(*αx*, *y*, *t*) = ⎧⎪⎨⎪⎩*Px*, *y*, *tα* , *i f α* > 0 *<sup>H</sup>*(*t*), *i f α* = 0 1 − *<sup>P</sup><sup>x</sup>*, *y*, *tα* , *i f α* < 0*; (FI5) <sup>P</sup>*(*<sup>x</sup>*, *z*, *t*) ∗ *<sup>P</sup>*(*y*, *<sup>z</sup>*,*<sup>s</sup>*) ≤ *<sup>P</sup>*(*x* + *y*, *z*, *t* + *s*), (∀)*<sup>t</sup>*,*<sup>s</sup>* > 0*; (FI6)* lim *t*→∞ *<sup>P</sup>*(*<sup>x</sup>*, *y*, *t*) = 1*.*

**Definition 10.** *[26] Let X be a linear space over* R*. A fuzzy inner product on X is a mapping* < ·, · >: *X* × *X* → F<sup>∗</sup>(R)*, where* F∗(R) = {*η* ∈ F(R) : *η*(*t*) = 0 *if t* < <sup>0</sup>}*, with the following properties* (∀)*<sup>x</sup>*, *y*, *z* ∈ *<sup>X</sup>*,(∀)*<sup>r</sup>* ∈ R*:*

*(FIP1)* < *x* + *y*, *z* >=< *x*, *z* > ⊕ < *y*, *z* >*; (FIP2)* < *rx*, *y* <sup>&</sup>gt;=| *r*˜ |< *x*, *y* >*; (FIP3)* < *x*, *y* >=< *y*, *x* >*; (FIP4) x* = 0 ⇒ < *x*, *x* > (*t*) = 0, (∀)*t* < 0*; (FIP5)* < *x*, *x* >= 0 ˜ *iff x* = 0*. Thepair*(*<sup>X</sup>*,<>)*iscalledfuzzyinnerproduct*

 ·, ·

#### **3. A New Approach for Fuzzy Inner Product Space**

We will denote by C the space of complex numbers and we will denote by <sup>R</sup><sup>∗</sup>+ the set of all strict positive real numbers.

 *space.*

**Definition 11.** *Let H be a linear space over* C*. A fuzzy set P in H* × *H* × C *is called a fuzzy inner product on H if it satisfies:*

*(FIP1) <sup>P</sup>*(*<sup>x</sup>*, *x*, *v*) = 0,(∀)*x* ∈ *<sup>H</sup>*,(∀)*<sup>v</sup>* ∈ C \ <sup>R</sup><sup>∗</sup>+*; (FIP2) <sup>P</sup>*(*<sup>x</sup>*, *x*, *t*) = 1,(∀)*t* ∈ <sup>R</sup><sup>∗</sup>+ *if and only if x* = 0*; (FIP3) <sup>P</sup>*(*αx*, *y*, *v*) = *Px*, *y*, *v*<sup>|</sup>*α*<sup>|</sup>,(∀)*<sup>x</sup>*, *y* ∈ *<sup>H</sup>*,(∀)*<sup>v</sup>* ∈ C,(∀)*<sup>α</sup>* ∈ C<sup>∗</sup>*; (FIP4) <sup>P</sup>*(*<sup>x</sup>*, *y*, *v*) = *<sup>P</sup>*(*y*, *x*, *<sup>v</sup>*),(∀)*<sup>x</sup>*, *y* ∈ *<sup>H</sup>*,(∀)*<sup>v</sup>* ∈ C*; (FIP5) <sup>P</sup>*(*x* + *y*, *z*, *v* + *w*) ≥ min{*P*(*<sup>x</sup>*, *z*, *<sup>v</sup>*), *<sup>P</sup>*(*y*, *z*, *<sup>w</sup>*)},(∀)*<sup>x</sup>*, *y*, *z* ∈ *<sup>H</sup>*,(∀)*<sup>v</sup>*, *w* ∈ C*; (FIP6) <sup>P</sup>*(*<sup>x</sup>*, *x*, ·) : R+ → [0, <sup>1</sup>],(∀)*<sup>x</sup>* ∈ *H is left continuous and* lim*t*→∞ *<sup>P</sup>*(*<sup>x</sup>*, *x*, *t*) = 1*; (FIP7) <sup>P</sup>*(*<sup>x</sup>*, *y*,*st*) ≥ min*P*(*<sup>x</sup>*, *<sup>x</sup>*,*s*<sup>2</sup>), *<sup>P</sup>*(*y*, *y*, *<sup>t</sup>*<sup>2</sup>),(∀)*<sup>x</sup>*, *y* ∈ *<sup>H</sup>*,(∀)*<sup>s</sup>*, *t* ∈ <sup>R</sup><sup>∗</sup>+*.*

*The pair* (*<sup>H</sup>*, *P*) *will be called fuzzy inner product space.*

**Example 1.** *Let H be a linear space over* C *and* < ·, · >: *H* × *H* → C *be an inner product. Then P* : *H* × *H* × C → [0, 1]*,*

$$P(\mathbf{x}, \mathbf{y}, \mathbf{s}) = \begin{cases} \frac{\mathbf{s}}{s + \|\mathbf{x}, \mathbf{y}, \mathbf{s}\|} \prime & \text{if } \mathbf{s} \in \mathbb{R}\_+^\* \\ 0, & \text{if } \mathbf{s} \in \mathbb{C} \backslash \mathbb{R}\_+^\* \end{cases}$$

*is a fuzzy inner product on H.*

Let verify now the conditions from the definition.

*(FIP1) <sup>P</sup>*(*<sup>x</sup>*, *x*, *v*) = 0,(∀)*x* ∈ *<sup>H</sup>*,(∀)*<sup>v</sup>* ∈ C \ <sup>R</sup><sup>∗</sup>+ is is obvious from definition of *P*. *(FIP2) <sup>P</sup>*(*<sup>x</sup>*, *x*, *t*) = 1,(∀)*v* ∈ <sup>R</sup><sup>∗</sup>+ ⇔ *t*+ |< *x*, *x* <sup>&</sup>gt;|= *<sup>t</sup>*,(∀)*<sup>t</sup>* > 0 ⇔|< *x*, *x* <sup>&</sup>gt;|= 0 ⇔ *x* = 0. *(FIP3) <sup>P</sup>*(*αx*, *y*, *v*) = *Px*, *y*, *v*<sup>|</sup>*α*<sup>|</sup>,(∀)*<sup>x</sup>*, *y* ∈ *<sup>H</sup>*,(∀)*<sup>v</sup>* ∈ C,(∀) *α* ∈ C is obvious for *v* ∈ C \ <sup>R</sup><sup>∗</sup>+. If*v*∈<sup>R</sup><sup>∗</sup>+,than

$$P(ax, y, \upsilon) = \frac{\upsilon}{\upsilon + |\{|} = \frac{\upsilon}{\upsilon + |a\mid \cdot| < \ge, y>|} = \frac{\frac{\upsilon}{|a|}}{\frac{\upsilon}{|a|} + |<\ge, y>|} = P\left(x, y, \frac{\upsilon}{|a|}\right).$$

*(FIP4) <sup>P</sup>*(*<sup>x</sup>*, *y*, *v*) = *<sup>P</sup>*(*y*, *x*, *<sup>v</sup>*),(∀)*<sup>x</sup>*, *y* ∈ *<sup>H</sup>*,(∀)*<sup>v</sup>* ∈ C is obvious for *v* ∈ C \ <sup>R</sup><sup>∗</sup>+.If *v* ∈ <sup>R</sup><sup>∗</sup>+, then *v* = *v*¯ and

$$P(\mathbf{x}, \mathbf{y}, \boldsymbol{\upsilon}) = \frac{\boldsymbol{\upsilon}}{\boldsymbol{\upsilon} + \, | < \, \mathbf{x}, \mathbf{y} > \boldsymbol{\upsilon} |} = \frac{\boldsymbol{\upsilon}}{\boldsymbol{\upsilon} + \, | < \, \mathbf{y}, \boldsymbol{\upsilon} > \boldsymbol{\upsilon} |} = P(\boldsymbol{y}, \mathbf{x}, \overline{\mathbf{y}})$$

*(FIP5) <sup>P</sup>*(*x* + *y*, *z*, *v* + *w*) ≥ min{*P*(*<sup>x</sup>*, *z*, *<sup>v</sup>*), *<sup>P</sup>*(*y*, *z*, *<sup>w</sup>*)},(∀)*<sup>x</sup>*, *y*, *z* ∈ *<sup>H</sup>*,(∀)*<sup>v</sup>*, *w* ∈ C. If at least one of *v* and *w* is from C \ <sup>R</sup><sup>∗</sup>+, then the result is obvious.

If *v*, *w* ∈ <sup>R</sup><sup>∗</sup>+, let us assume without loss of generality that *<sup>P</sup>*(*<sup>x</sup>*, *z*, *v*) ≤ *<sup>P</sup>*(*y*, *z*, *<sup>w</sup>*). Then

*v v*+ |< *x*, *z* >| ≤ *w w*+ |< *y*, *z* >| ⇒ ⇒ *v*+ |< *x*, *z* >| *v* ≥ *w*+ |< *y*, *z* >| *w* ⇒ ⇒ 1 + |< *x*, *z* >| *v* ≥ 1 + |< *y*, *z* >| *w* ⇒ ⇒ |< *x*, *z* >| *v* ≥ |< *y*, *z* >| *w* ⇒ ⇒ *w v* |< *x*, *z* >|≥|< *y*, *z* >|⇒ ⇒|< *x*, *z* >| +*wv* |< *x*, *z* >|≥|< *x*, *z* >| + |< *y*, *z* >|⇒ ⇒ *v* + *w v* |< *x*, *z* >|≥|< *x* + *y*, *z* >|⇒ ⇒ |< *x*, *z* >| *v* ≥ |< *x* + *y*, *z* >| *v* + *w* ⇒ ⇒ |< *x*, *z* >| *v* + 1 ≥ |< *x* + *y*, *z* >| *v* + *w* + 1 ⇒ ⇒ *v*+ |< *x*, *z* >| *v* ≥ (*v* + *w*)+ |< *x* + *y*, *z* >| *v* + *w* ⇒ ⇒ *v v*+ |< *x*, *z* >| ≤ *v* + *w* (*v* + *w*)+ |< *x* + *y*, *z* >| ⇒ ⇒ *<sup>P</sup>*(*<sup>x</sup>*, *z*, *v*) ≤ *<sup>P</sup>*(*x* + *y*, *z*, *v* + *w*)

⇒ *<sup>P</sup>*(*x* + *y*, *z*, *v* + *w*) ≥ min{*P*(*<sup>x</sup>*, *z*, *<sup>v</sup>*), *<sup>P</sup>*(*y*, *z*, *<sup>w</sup>*)},(∀)*<sup>x</sup>*, *y*, *z* ∈ *<sup>H</sup>*,(∀)*<sup>v</sup>*, *w* ∈ C.

*(FIP6) <sup>P</sup>*(*<sup>x</sup>*, *x*, ·) : R+ → [0, <sup>1</sup>],(∀)*<sup>x</sup>* ∈ *H* is left continuous function and lim*t*→∞ *<sup>P</sup>*(*<sup>x</sup>*, *x*, *t*) = 1.

$$\lim\_{t \to \infty} P(\mathbf{x}, \mathbf{x}, t) = \lim\_{t \to \infty} \frac{t}{t + |\le \mathbf{x}, \mathbf{x} > |} = \lim\_{t \to \infty} \frac{t}{t(1 + \frac{|\le x, \mathbf{x} > |}{t})} = 1.1$$

*<sup>F</sup>*(*<sup>x</sup>*, *x*, ·) is left continuous in *t* > 0 follows from definition.

*(FIP7) <sup>P</sup>*(*<sup>x</sup>*, *y*,*st*) ≥ min{*P*(*<sup>x</sup>*, *<sup>x</sup>*,*s*<sup>2</sup>), *<sup>P</sup>*(*y*, *y*, *<sup>t</sup>*<sup>2</sup>)},(∀)*<sup>x</sup>*, *y* ∈ *<sup>H</sup>*,(∀)*<sup>s</sup>*, *t* ∈ <sup>R</sup><sup>∗</sup>+. If at least one of *s* and *t* is from C \ <sup>R</sup><sup>∗</sup>+, then the result is obvious.

If *s*, *t* ∈ <sup>R</sup><sup>∗</sup>+, let us assume without loss of generality that *<sup>P</sup>*(*<sup>x</sup>*, *<sup>x</sup>*,*s*<sup>2</sup>) ≤ *<sup>P</sup>*(*y*, *y*, *<sup>t</sup>*<sup>2</sup>). Then

$$\frac{s^2}{s^2 + ||} \le \frac{t^2}{t^2 + ||} \Leftrightarrow$$

$$t^2 \mid | \ge s^2 \mid |\ .$$

Thus by Cauchy–Schwartz inequality we obtain

$$s \mid < x, y> \mid \le \sqrt{||} \cdot s\sqrt{||} \le \sqrt{||} \cdot t\sqrt{||} = t \mid < x, x>| \Rightarrow t \mid \le x$$

$$\Rightarrow s^2 \mid < x, y> \mid \le st \mid < x, x>| \Rightarrow$$

$$\Rightarrow s^3 t + s^2 \mid < x, y>| \le s^3 t + st \mid < x, x>| \Rightarrow$$

$$\Rightarrow s^2 (st + ||) \le st(s^2 + ||) \Rightarrow$$

$$\frac{s^2}{s^2 + ||} \le \frac{st}{st + ||} \Rightarrow$$

$$\Rightarrow P(x, x, s^2) \le P(x, y, st) \Rightarrow$$

$$\Rightarrow P(x, y, st) \ge \min\left\{P(x, x, s^2), P(y, y, t^2)\right\}. \text{ (\%)} \forall x, y \in H\_r(\forall) \text{s}, t \in \mathbb{R}\_+^\*.$$

**Proposition 1.** *For x*, *y* ∈ *H*, *v* ∈ C *and α* ∈ C *we have*

$$P(\mathbf{x}, \mathbf{a}y, \upsilon) = P(\mathbf{x}, y, \frac{\upsilon}{|\alpha|}).$$

**Proof.** From (FIP3) and (FIP4) it follows *<sup>P</sup>*(*<sup>x</sup>*, *<sup>α</sup>y*, *v*) = *<sup>P</sup>*(*αy*, *x*, *v*) = *Py*, *x*, *v*|*α*|= *Px*, *y*, *v*|*α*| = *Px*, *y*, *v*|*α*|.

**Proposition 2.** *For x* ∈ *H*, *v* ∈ <sup>R</sup><sup>∗</sup>+ *we have*

$$P(x,0,v) = 1.$$

**Proof.** From (FIP3) and (FIP6) it follows

$$P(\mathbf{x},0,\upsilon) = P(\mathbf{x},0,2n\upsilon) = P(\mathbf{x},\mathbf{x}-\mathbf{x},n\upsilon+n\upsilon) \ge \min\{P(\mathbf{x},\mathbf{x},n\upsilon), P(\mathbf{x},\mathbf{x},n\upsilon)\} = 1$$

$$P(\mathbf{x},\mathbf{x},n\upsilon) \stackrel{\upsilon \to \infty}{\longrightarrow} 1$$

So *<sup>P</sup>*(*<sup>x</sup>*, 0, *v*) = 1.

**Proposition 3.** *For y* ∈ *H*, *v* ∈ <sup>R</sup><sup>∗</sup>+ *we have*

$$P(0, y, v) = 1.$$

**Proposition 4.** *<sup>P</sup>*(*<sup>x</sup>*, *y*, ·) : R+ → [0, 1] *is a monotonic non-decreasing function on* R+, (∀)*<sup>x</sup>*, *y* ∈ *H.*

**Proof.** Let *s*, *t* ∈ R+, *s* ≤ *t*. Then (∃)*p* such that *t* = *s* + *p* and

$$P(\mathbf{x}, y, \mathbf{t}) = P(\mathbf{x} + \mathbf{0}, y, \mathbf{s} + \mathbf{p}) \ge \min\{P(\mathbf{x}, y, \mathbf{s}), P(\mathbf{0}, y, \mathbf{p})\} = P(\mathbf{x}, y, \mathbf{s}).$$

Hence *<sup>P</sup>*(*<sup>x</sup>*, *y*,*<sup>s</sup>*) ≤ *<sup>P</sup>*(*<sup>x</sup>*, *y*, *t*) for *s* ≤ *t*.

**Corollary 1.** *<sup>P</sup>*(*<sup>x</sup>*, *y*,*st*) ≥ min*P*(*<sup>x</sup>*, *<sup>y</sup>*,*s*<sup>2</sup>), *<sup>P</sup>*(*<sup>x</sup>*, *y*, *<sup>t</sup>*<sup>2</sup>),(∀)*<sup>x</sup>*, *y* ∈ *<sup>H</sup>*,(∀)*<sup>s</sup>*, *t* ∈ <sup>R</sup><sup>∗</sup>+*.*

**Proof.** Let *s*, *t* ∈ R+, *s* ≤ *t*. Then

$$P(\mathfrak{x}, \mathfrak{y}, \mathfrak{s}^2) \le P(\mathfrak{x}, \mathfrak{y}, \mathfrak{st}) \le P(\mathfrak{x}, \mathfrak{y}, \mathfrak{t}^2).$$

Hence *<sup>P</sup>*(*<sup>x</sup>*, *y*,*st*) ≥ min*P*(*<sup>x</sup>*, *<sup>y</sup>*,*s*<sup>2</sup>), *<sup>P</sup>*(*<sup>x</sup>*, *y*, *<sup>t</sup>*<sup>2</sup>). Let now *s*, *t* ∈ R+, *t* ≤ *s*. Then

$$P(\mathbf{x}, \mathbf{y}, t^2) \le P(\mathbf{x}, \mathbf{y}, \text{st}) \le P(\mathbf{x}, \mathbf{y}, \mathbf{s}^2).$$

Hence *<sup>P</sup>*(*<sup>x</sup>*, *y*,*st*) ≥ min*P*(*<sup>x</sup>*, *<sup>y</sup>*,*s*<sup>2</sup>), *<sup>P</sup>*(*<sup>x</sup>*, *y*, *<sup>t</sup>*<sup>2</sup>).

**Proposition 5.** *<sup>P</sup>*(*<sup>x</sup>*, *y*, *v*) ≥ min{*P*(*<sup>x</sup>*, *y* − *z*, *<sup>v</sup>*), *<sup>P</sup>*(*<sup>x</sup>*, *y* + *z*, *<sup>v</sup>*)},(∀)*<sup>x</sup>*, *y*, *z* ∈ *<sup>H</sup>*,(∀)*<sup>v</sup>* ∈ C*.*

**Proof.**

$$P(\mathbf{x}, y, v) = P(\mathbf{x}, 2y, 2v) = P(\mathbf{x}, y + z + y - z, v + v) \ge \min\{P(\mathbf{x}, y + z, v), P(\mathbf{x}, y - z, v)\}.$$

$$\text{Hence } P(\mathbf{x}, y, v) \ge \min\{P(\mathbf{x}, y - z, v), P(\mathbf{x}, y + z, v)\}.\quad \square$$

**Theorem 1.** *If* (*<sup>H</sup>*, *P*) *be a fuzzy inner product space, then N* : *X* × [0, ∞) → [0, 1] *defined by*

$$N(\mathbf{x}, t) = P(\mathbf{x}, \mathbf{x}, t^2)$$

*is a fuzzy norm on X.*

**Proof.** *(N1) <sup>N</sup>*(*<sup>x</sup>*, 0) = *<sup>P</sup>*(*<sup>x</sup>*, *x*, 0) = 0,(∀)*x* ∈ *H* from **(FIP1)**; *(N2)* [*N*(*<sup>x</sup>*, *t*) = 1,(∀)*t* > 0] ⇔ *<sup>P</sup>*(*<sup>x</sup>*, *x*, *t*2) = 1,(∀)*t* > 0 ⇔ *x* = 0 from **(FIP2)**; *(N3) <sup>N</sup>*(*<sup>λ</sup><sup>x</sup>*, *t*) = *<sup>P</sup>*(*<sup>λ</sup><sup>x</sup>*, *λ<sup>x</sup>*, *t*2) = *Px*, *λ<sup>x</sup>*, *t*2 |*λ*| = *Pλ<sup>x</sup>*, *x*, *t*¯2 |*λ*| = *Pλ<sup>x</sup>*, *x*, *t*2 |*λ*| = *Px*, *x*, *t*2 |*λ*|<sup>2</sup> = = *Nx*, *t*<sup>|</sup>*λ*<sup>|</sup>, (∀)*t* ≥ 0,(∀)*λ* ∈ K<sup>∗</sup>; *(N4) <sup>N</sup>*(*x* + *t*, *t* + *s*) ≥ min{*N*(*<sup>x</sup>*, *t*), *<sup>N</sup>*(*y*,*<sup>s</sup>*)},(∀)*<sup>x</sup>*, *y* ∈ *<sup>H</sup>*,(∀)*<sup>t</sup>*,*<sup>s</sup>* ≥ 0. If *t* = 0 or *s* = 0 the previous inequality is obvious. We asume that *t*,*<sup>s</sup>* > 0.

$$\begin{array}{rcl}N(\mathbf{x}+\mathbf{y},t+\mathbf{s})&=&P\left(\mathbf{x}+\mathbf{y},\mathbf{x}+\mathbf{y},(t+\mathbf{s})^{2}\right)=\\&=&P(\mathbf{x}+\mathbf{y},\mathbf{x}+\mathbf{y},t^{2}+\mathbf{s}^{2}+\mathbf{ts}+\mathbf{ts})\geq\\&\geq&P(\mathbf{x},\mathbf{x}+\mathbf{y},t^{2}+\mathbf{ts})\land P(\mathbf{y},\mathbf{x}+\mathbf{y},\mathbf{s}^{2}+\mathbf{ts})\geq\\&\geq&P(\mathbf{x},\mathbf{x},t^{2})\land P(\mathbf{x},\mathbf{y},\mathbf{ts})\land P(\mathbf{y},\mathbf{x},\mathbf{ts})\land P(\mathbf{y},\mathbf{y},\mathbf{s}^{2})=\\&=&P(\mathbf{x},\mathbf{x},t^{2})\land P(\mathbf{y},\mathbf{y},\mathbf{s}^{2})=\\&=&\min\{N(\mathbf{x},\mathbf{t}),N(\mathbf{y},\mathbf{s})\};\end{array}$$

*(N5)* From **(FIP6)** it result that *<sup>N</sup>*(*<sup>x</sup>*, ·) is left continuous and lim*t*→∞ *<sup>N</sup>*(*<sup>x</sup>*, *t*) = 1.

#### **4. Conclusions and Future Works**

In this paper, we wrote a literature review regarding the diverse approaches of fuzzy inner product space concept, but we also introduced a new approach.

We have thus laid the ground for further research on the problems within the fuzzy Hilbert space theory, searching for analogies in this fuzzy context for the Pythagorean theorem, for the parallelogram law, as well as for other orthogonality problems.

The following step would be to study the linear and bounded operators in a fuzzy Hilbert space. Recently, there have been many important results concerning the linear and bounded operators in a fuzzy Banach space (see [30–34]), fact which motivates us even further to try to achieve this goal.

This research will be followed by other papers in which we will firstly define the concept of an adjoint of a linear and bounded operator on a fuzzy Hilbert space. This concept will allow us to study important classes of operators such as self-adjoint operators, normal operators and unitary operators. We will then follow up with the spectral theory and we will also construct a analytic functional calculus.

Last, but not least, we will study the aforementioned orthogonality in the fuzzy Hilbert space. Thus, this will enable us to observe the properties of the self-adjoint projections and to undertake directly decompositions of the fuzzy Hilbert space.

This paper summarized the current research status of the fuzzy inner product spaces and can thus facilitate researchers in writing their future papers on this topic.

**Author Contributions:** Conceptualization, L.P.; Data curation, L.S.; Formal analysis, L.S. Funding acquisition, L.P.; Resources, L.S.; Writing–review and editing, L.P. Both authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The authors are extremely grateful to the editor and reviewers, for their very carefully reading of the paper and for their valuable comments and suggestions which have been useful to increase the scientific quality and presentation of the paper. The authors wish to record their sincere gratitude to S. N ˘ad ˘aban for his help in the preparation of this paper.

**Conflicts of Interest:** The authors declare no conflict of interest.
