In this section, at first we give the concept of linear 2normed space and later the concept of 2fuzzy 2normed linear space and its fundamental properties with help of [8]. For more details, we refer the readers to [7, 8, 28, 29].
Definition 2.1 [28]
Let X be a real vector space of dimension greater than 1 and let \parallel \u2022,\u2022\parallel be a realvalued function on X\times X satisfying the following four properties:

(1)
\parallel x,y\parallel =0 if and only if x and y are linearly dependent,

(2)
\parallel x,y\parallel =\parallel y,x\parallel,

(3)
\parallel x,\alpha y\parallel =\alpha \parallel x,y\parallel for any \alpha \in \mathbb{R},

(4)
\parallel x,y+z\parallel \le \parallel x,y\parallel +\parallel x,z\parallel,
\parallel \u2022,\u2022\parallel is called a 2norm on X and the pair (X,\parallel \u2022,\u2022\parallel ) is called a linear 2normed space.
Definition 2.2 [7]
Let X be a linear space over S (a field of real or complex numbers). A fuzzy subset N of X\times \mathbb{R} (ℝ, the set of real numbers) is called a fuzzy norm on X if and only if:
(N1) For all t\in \mathbb{R} with t\le 0, N(x,t)=0,
(N2) For all t\in \mathbb{R} with t>0, N(x,t)=1 if and only if x=0,
(N3) For all t\in \mathbb{R} with t>0, N(\lambda x,t)=N(x,\frac{t}{\lambda }), if \lambda \ne 0, \lambda \in S,
(N4) For all s,t\in \mathbb{R}, x,y\in X, N(x+y,s+t)\ge min\{N(x,s),N(y,t)\},
(N5) N(x,\cdot ) is a nondecreasing function of t\in \mathbb{R} and {lim}_{t\to \mathrm{\infty}}N(x,t)=1.
Then (X,N) is called a fuzzy normed linear space or, in short, fNLS.
Theorem 2.1 [7]
Let (X,N) be an fNLS. Assume the condition that
(N6) N(x,t)>0 for all t>0 implies x=0.
Define {\parallel x\parallel}_{\alpha}=inf\{t:N(x,t)\ge \alpha \}, \alpha \in (0,1). Then \{{\parallel \u2022\parallel}_{\alpha}:\alpha \in (0,1)\} is an ascending family of norms on X. We call these norms αnorms on X corresponding to the fuzzy norm on X.
Definition 2.3 Let X be any nonempty set and \mathrm{\Im}(X) be the set of all fuzzy sets on X. For U,V\in \mathrm{\Im}(X) and \lambda \in S the field of real numbers, define
U+V=\{(x+y,\nu \wedge \mu ):(x,\nu )\in U,(y,\mu )\in V\}
and \lambda U=\{(\lambda x,\nu ):(x,\nu )\in U\}.
Definition 2.4 A fuzzy linear space \stackrel{\u02c6}{X}=X\times (0,1] over the number field S, where the addition and scalar multiplication operation on X are defined by (x,\nu )+(y,\mu )=(x+y,\nu \wedge \mu ), \lambda (x,\nu )=(\lambda x,\nu ) is a fuzzy normed space if to every (x,\nu )\in \stackrel{\u02c6}{X}, there is associated a nonnegative real number, \parallel (x,\nu )\parallel, called the fuzzy norm of (x,\nu ), in such a way that

(i)
\parallel (x,\nu )\parallel =0 iff x=0 the zero element of X, \nu \in (0,1],

(ii)
\parallel \lambda (x,\nu )\parallel =\lambda \parallel (x,\nu )\parallel for all (x,\nu )\in \stackrel{\u02c6}{X} and all \lambda \in S,

(iii)
\parallel (x,\nu )+(y,\mu )\parallel \le \parallel (x,\nu \wedge \mu )\parallel +\parallel (y,\nu \wedge \mu )\parallel for all (x,\nu ),(y,\mu )\in \stackrel{\u02c6}{X},

(iv)
\parallel (x,{\bigvee}_{t}{\nu}_{t})\parallel ={\bigwedge}_{t}\parallel (x,{\nu}_{t})\parallel for all {\nu}_{t}\in (0,1].
Definition 2.5 [8]
Let X be a nonempty and \mathrm{\Im}(X) be the set of all fuzzy sets in X. If f\in \mathrm{\Im}(X), then f=\{(x,\mu ):x\in X\text{and}\mu \in (0,1]\}. Clearly, f is a bounded function for f(x)\le 1. Let S be the space of real numbers, then \mathrm{\Im}(X) is a linear space over the field S where the addition and multiplication are defined by
f+g=\{(x,\mu )+(y,\eta )\}=\{(x+y,\mu \wedge \eta ):(x,\mu )\in f\text{and}(y,\eta )\in g\}
and
\lambda f=\{(\lambda x,\mu ):(x,\mu )\in f\},
where \lambda \in S.
The linear space \mathrm{\Im}(X) is said to be a normed space if for every f\in \mathrm{\Im}(X), there is associated a nonnegative real number \parallel f\parallel called the norm of f in such a way that

(i)
\parallel f\parallel =0 if and only if f=0. For

(ii)
\parallel \lambda f\parallel =\lambda \parallel f\parallel, \lambda \in S. For
\begin{array}{rcl}\parallel \lambda f\parallel & =& \{\parallel \lambda (x,\mu )\parallel :(x,\mu )\in f,\lambda \in S\}\\ =& \{\lambda \parallel (x,\mu )\parallel :(x,\mu )\in f\}=\lambda \parallel f\parallel .\end{array}

(iii)
\parallel f+g\parallel \le \parallel f\parallel +\parallel g\parallel for every f,g\in \mathrm{\Im}(X). For
\begin{array}{rcl}\parallel f+g\parallel & =& \{\parallel (x,\mu )+(y,\eta )\parallel :x,y\in X,\mu ,\eta \in (0,1]\}\\ =& \{\parallel (x+y),(\mu \wedge \eta )\parallel :x,y\in X,\mu ,\eta \in (0,1]\}\\ =& \{\parallel (x,\mu \wedge \eta )\parallel +\parallel (y,\mu \wedge \eta )\parallel :(x,\mu )\in f,(y,\eta )\in g\}\\ =& \parallel f\parallel +\parallel g\parallel .\end{array}
Then (\mathrm{\Im}(X),\parallel \u2022\parallel ) is a normed linear space.
Definition 2.6 [8]
A 2fuzzy set on X is a fuzzy set on \mathrm{\Im}(X).
Definition 2.7 [8]
Let \mathrm{\Im}(X) be a linear space over the real field S. A fuzzy subset N of \mathrm{\Im}(X)\times \mathrm{\Im}(X)\times \mathbb{R} (ℝ, a set of real numbers) is called a 2fuzzy 2norm on X (or a fuzzy 2norm on \mathrm{\Im}(X)) if and only if
(2N1) for all t\in \mathbb{R} with t\le 0, N({f}_{1},{f}_{2},t)=0,
(2N2) for all t\in \mathbb{R} with t>0, N({f}_{1},{f}_{2},t)=1 if and only if {f}_{1} and {f}_{2} are linearly dependent,
(2N3) N({f}_{1},{f}_{2},t) is invariant under any permutation of {f}_{1}, {f}_{2},
(2N4) for all t\in \mathbb{R} with t>0, N({f}_{1},\lambda {f}_{2},t)=N({f}_{1},{f}_{2},\frac{t}{\lambda }), if \lambda \ne 0, \lambda \in S,
(2N5) for all s,t\in \mathbb{R},
N({f}_{1},{f}_{2}+{f}_{3},s+t)\ge min\{N({f}_{1},{f}_{2},s),N({f}_{1},{f}_{3},t)\},
(2N6) N({f}_{1},{f}_{2},\cdot ):(0,\mathrm{\infty})\to [0,1] is continuous,
(2N7) {lim}_{t\to \mathrm{\infty}}N({f}_{1},{f}_{2},t)=1.
Then (\mathrm{\Im}(X),N) is a fuzzy 2normed linear space or (X,N) is a 2fuzzy 2normed linear space.
Remark 2.1 In a 2fuzzy 2normed linear space (X,N), N({f}_{1},{f}_{2},\cdot ) is a nondecreasing function of ℝ for all {f}_{1},{f}_{2}\in \mathrm{\Im}(X).
Theorem 2.2 [8]
Let (\mathrm{\Im}(X),N) be a fuzzy 2normed linear space. Assume that
(2N8) N({f}_{1},{f}_{2},t)>0 for all t>0 implies {f}_{1} and {f}_{2} are linearly dependent.
Define {\parallel {f}_{1},{f}_{2}\parallel}_{\alpha}=inf\{t:N({f}_{1},{f}_{2}t)\ge \alpha ,\alpha \in (0,1)\}.
Then \{{\parallel \u2022,\u2022\parallel}_{\alpha}:\alpha \in (0,1)\} is an ascending family of 2norms on \mathrm{\Im}(X). These 2norms are called α2norms on \mathrm{\Im}(X) corresponding to the 2fuzzy 2norm on X.