In this section, at first we give the concept of linear 2-normed space and later the concept of 2-fuzzy 2-normed linear space and its fundamental properties with help of . For more details, we refer the readers to [7, 8, 28, 29].
Definition 2.1 
Let X be a real vector space of dimension greater than 1 and let be a real-valued function on satisfying the following four properties:
if and only if x and y are linearly dependent,
for any ,
is called a 2-norm on X and the pair is called a linear 2-normed space.
Definition 2.2 
Let X be a linear space over S (a field of real or complex numbers). A fuzzy subset N of (ℝ, the set of real numbers) is called a fuzzy norm on X if and only if:
(N1) For all with , ,
(N2) For all with , if and only if ,
(N3) For all with , , if , ,
(N4) For all , , ,
(N5) is a non-decreasing function of and .
Then is called a fuzzy normed linear space or, in short, f-NLS.
Theorem 2.1 
Let be an f-NLS. Assume the condition that
(N6) for all implies .
Define , . Then is an ascending family of norms on X. We call these norms α-norms on X corresponding to the fuzzy norm on X.
Definition 2.3 Let X be any non-empty set and be the set of all fuzzy sets on X. For and the field of real numbers, define
Definition 2.4 A fuzzy linear space over the number field S, where the addition and scalar multiplication operation on X are defined by , is a fuzzy normed space if to every , there is associated a non-negative real number, , called the fuzzy norm of , in such a way that
iff the zero element of X, ,
for all and all ,
for all ,
for all .
Definition 2.5 
Let X be a non-empty and be the set of all fuzzy sets in X. If , then . Clearly, f is a bounded function for . Let S be the space of real numbers, then is a linear space over the field S where the addition and multiplication are defined by
The linear space is said to be a normed space if for every , there is associated a non-negative real number called the norm of f in such a way that
if and only if . For
, . For
for every . For
Then is a normed linear space.
Definition 2.6 
A 2-fuzzy set on X is a fuzzy set on .
Definition 2.7 
Let be a linear space over the real field S. A fuzzy subset N of (ℝ, a set of real numbers) is called a 2-fuzzy 2-norm on X (or a fuzzy 2-norm on ) if and only if
(2-N1) for all with , ,
(2-N2) for all with , if and only if and are linearly dependent,
(2-N3) is invariant under any permutation of , ,
(2-N4) for all with , , if , ,
(2-N5) for all ,
(2-N6) is continuous,
Then is a fuzzy 2-normed linear space or is a 2-fuzzy 2-normed linear space.
Remark 2.1 In a 2-fuzzy 2-normed linear space , is a non-decreasing function of ℝ for all .
Theorem 2.2 
Let be a fuzzy 2-normed linear space. Assume that
(2-N8) for all implies and are linearly dependent.
Then is an ascending family of 2-norms on . These 2-norms are called α-2-norms on corresponding to the 2-fuzzy 2-norm on X.