EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 4, Article Number 6541
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Completeness and Compactness on Hesitant Fuzzy
Normed Linear Spaces

Krishnamoorthy Kavitha1, Prakasam Muralikrishna2,∗

1 PG and Research Department of Mathematics, Muthurangam Government Arts College
(Autonomous) (Affiliated to Thiruvalluvar University, Serkkadu, Vellore), Vellore-632002,
Tamil Nadu, India
2 PG and Research Department of Mathematics, Muthurangam Government Arts College
(Autonomous) (Affiliated to Thiruvalluvar University, Serkkadu, Vellore), Vellore-632002,
Tamil Nadu, India

Abstract. In this work, we examine the properties of completeness and compactness on hesitant
fuzzy normed linear space, also address the same properties on intuitionistic hesitant fuzzy normed
linear space in finite dimension using definitions, lemmas, and theorems. We also investigate the
continuity of underlying t-norms and co-t-norm on finite-dimensional intuitionistic hesitant fuzzy
normed linear space.

2020 Mathematics Subject Classifications: 00A05, 00A22

Key Words and Phrases: Hesitant fuzzy, intuitionistic hesitant fuzzy, normed linear space

1. Introduction

In 1965, Zadeh [1] invented fuzzy set theory. The search for fuzzy equivalents of clas-
sical theories has been intense since Zadeh’s groundbreaking work. Additionally, other
areas, fuzzy metric spaces along with fuzzy normed linear spaces have seen advancements,
In [[2],[3]] Two kinds of fuzzy bounded linear operators—strong and weak—are developed
in this study, along with the concept regarding boundedness of a linear operator out of one
fuzzy normed linear space to another fuzzy normed linear space. A relationship between
fuzzy boundedness and fuzzy continuity is examined. The concepts of fuzzy dual spaces
and fuzzy bounded linear functionals are defined, establish Uniform Boundedness Princi-
ple, Closed Graph , Open Mapping and the Hahn-Banach Theorem, In [4] a fuzzy normed
linear space, the terms ”strongly and weakly fuzzy convergent sequence,” are defined over
this study. Fixed point theorems for fuzzy non-expansive mappings are established, along
with the notions of uniformly convex fuzzy normed linear space, fuzzy normal structure,

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6541

Email addresses: kavithamths@gmail.com (K. Kavitha),
pmkrishna@rocketmail.com (P. Muralikrishna)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



K. Kavitha, P. Muralikrishna / Eur. J. Pure Appl. Math, 18 (4) (2025), 6541 2 of 14

and fuzzy non-expansive mapping.
In [5] an introduction to fuzzy normed linear space is given. It has been demonstrated that
fuzzy norms are equivalent up to fuzzy equivalency in a finite dimensional fuzzy normed
linear space. It is demonstrated that fuzzy subspaces of a fuzzy normed linear space among
finite dimensions must be full fuzzy normed linear spaces, [6] present the idea of a fuzzy
metric space in this study. In a fuzzy metric space, The separation among two points, is
a normal, convex, upper semicontinuous, non-negative fuzzy number. A few fixed point
theorems are proved and the properties of fuzzy metric spaces are examined.
In [7] a few fuzzy topological vector space properties are examined. Additionally, for a
fuzzy linear topology, necessary and sufficient criteria are shown for a family of fuzzy sets
in vector space E to be the family of all neighborhoods of zero.As a generalized fuzzy
set, Atanassov [8] developed the idea of intuitionistic fuzzy sets. Originating the concept
of intuitionistic fuzzy metric space was J.H. Park [9] and researched a few fundamental
characteristics. However, a significant addition to intuitionistic fuzzy topological spaces is
made by Saadati Park [9]. They have also examined certain fundamental characteristics
in intuitionistic fuzzy normed linear spaces and introduced the idea of such spaces. Many
studies have been conducted on intuitive fuzzy sets, including those by T.K. Mandal and
S.K. Samanta [10],[11].[12] N. Thillaigovindan et al. Vijayabalaji et al. [13] recently ob-
tained some results and presented the idea of intuitionistic fuzzy n-normed linear space.
Bag et al., [10], introduced the concept of a fuzzy normed linear space, which T.K. Samanta
et al. [14] examined. They stated an intuitionistic fuzzy normed linear space through a
general context (using the t-norm ∗ along with the t-co-norm ⋄ correspondingly).
In finite dimensional intuitionistic fuzzy normed linear space, they mostly examined vari-
ous outcomes. However, their findings rely on the intuitionistic fuzzy norm’s decomposition
theorem as part of a family of crisp norm pairings, because they have added requirements
on the t-norm and t-conorm as a1 ∗ a1 = a1 and a1 ⋄ a1 = a1, ∀a1 ∈ 0, 1 , leading to
∗ = min and ⋄ = max. The abstraction of the t-norm and t-conorm is thus practically
lost. However, certain of the requirements involving the functions N(x1, t1) and M(x1, t1)
in the definition taken into consideration , the relation N(x1, t1) + M(x1, t1) ≤ 1. The
following describes how the paper is structured: Section 2 describes, some preliminary
the outcomes were presented , notion of hesitant fuzzy and intuitionistic hesitant fuzzy
normed linear spaces. Section 3 and section 4 demonstrates some fundamental findings
about completeness and compactness are demonstrated in finite dimension hesitant fuzzy
normed linear space along with intuitionistic hesitant fuzzy normed linear space.

2. Preliminaries

This section provides pre-existing definitions of fuzzy sets, including some fundamental
notions.

Definition 1. [15]
∗ : [0, 1] × [0, 1] → [0, 1] in binary exists t−norm in case it meets these requirements

listed below:



K. Kavitha, P. Muralikrishna / Eur. J. Pure Appl. Math, 18 (4) (2025), 6541 3 of 14

[a] ∗ is associative and commutative.
[b] a11 ∗ 1 = a11, ∀ a11 ∈ [0, 1].
[c] a11∗b11 ≤ c11∗d11 whenever a11 ≤ c11 and b11 ≤ d11 for each a11, b11, c11, d11 ∈ [0, 1].

It can be described as the continuous t−norm if ∗ is continuous.

Definition 2. [15]
An operation that is binary ⋄ : [0, 1]× [0, 1] → [0, 1] is a t−co-norm when it meets the

requirements listed below:
[a] ⋄ is associative and commutative.
[b] a11 ⋄ 1 = a11, ∀ a11 ∈ [0, 1].
[c] a11⋄b11 ≤ c11⋄d11 whenever a11 ≤ c11 and b11 ≤ d11 for each a11, b11, c11, d11 ∈ [0, 1].
When ⋄ is continuous, it has to be referred as continuous t−co-norm.

Definition 3. [16] Consider the nonempty set V combined with algebraic operations +, �
fulfil, (V,+) is a group and regarding scalar multiplication,

(i) k1(a1 + b1) = k1a1 + k1b1

(ii) (k1 + l1)a1 = k1a1 +1 k1b

(iii) k1(la1) = (k1l1)a1

(iv) 1.a1 = a1, ∀a1, b1, c1 ∈ V and k1, l1 ∈ R∗

The triple (V,+, �) is referred to as vector space.

Definition 4. [17] The fuzzy set N in X× [0,∞) over a linear space X constitutes a fuzzy
norm over X when it meets this criteria,

(i) (FN1) N(x1, 0) = 0, ∀x1 ∈ X

(ii) (FN2) N(x1, t) = 1, ∀t > 0 iff x1 = 0.

(iii) (FN3)N(λx1, t) = N
(
x1,

t
|λ|

)
, ∀ x1 ∈ X,∀ t > 0, ∀ λ ∈ K∗ ,(K∗ is non negative real

numbers)

(iv) (FN4)N(x1 + y1, t+ s) ≥ N(x1, t) ∗N(y1, s), ∀ x1, y1 ∈ X,∀ t, s > 0.

(v) (FN5) ∀x1 ∈ X, N(x1, •) is left continuous along with limt→∞N(x1, t) = 1.

Thus,(X,N, ∗) known as fuzzy normed linear space.

Definition 5. [16] Hesitant Fuzzy Normed Linear Space : Given a vector space V
through this field F, ∗ consists of t-norm, together with H : V × [0,∞) → P [0, 1] exists as
a hesitant fuzzy set with the subsequent characteristics, t1, t2 > 0 and ∀ x, y ∈ V

(i) H(x, 0) = ∅∗ (Empty set), ∀ x ∈ V.

(ii) H(x, t) = U∗ (Full set), ∀t > 0 iff x = 0.



K. Kavitha, P. Muralikrishna / Eur. J. Pure Appl. Math, 18 (4) (2025), 6541 4 of 14

(iii) H(µx, t) = H(x, t
|µ|), ∀x ∈ V,∀t ≥ 0, ∀µ ∈ R∗.

(iv) H(x + y, t1 + t2) ⊇ H(x, t1) ∩H(y, t2), ∀x, y ∈ V, ∀ t1, t2 ≥ 0.

(v) limt→∞H(x, t) = U∗.

Definition 6. [16] A sequence vn with a hesitant fuzzy normed linear space (V,H), known
as converges towards v ∈ V suppose every S∗ ̸= ∅∗ and t > 0,we could locate N using
H(vn − v, t) ⊃ U∗\S∗ ∀ n ≥ N. (or) limn→∞H(vn − v, t) = U∗.

Definition 7. [16] A sequence vn in a hesitant fuzzy normed space (V,H, ∗) is said to be
a cauchy sequence if for all ∅∗ ⊂ S∗ ⊂ U∗, t > 0 there is number N with H(vm − vn, t) ⊃
U∗\S∗ for all m,n ≥ N. (or) limn→∞H(vn − v, t) = U∗.

3. Finite Dimensional Hesitant Fuzzy Normed Linear Space

Finite dimensional hesitant fuzzy normed linear spaces are defined, their completeness,
and their compactness are examined in this section.

Definition 8. Let (V,H, ∗) be a hesitant fuzzy normed linear space. If dimension of a
vector space V is finite then it is called finite dimensional hesitant fuzzy normed linear
space.

Definition 9. Given a hesitant fuzzy normed space (V,H, ∗) as well as a subset W of V,
W signifies the closure of W, which corresponds to

⋂
{W ⊆ B : B is closed in V }.

Lemma 1. Consider (V,H, ∗) to become hesitant fuzzy normed space, along with W exists
as a subset of V. A sequence {wn} within W converges towards y,if and only if y ∈ w.

Definition 10. The completeness of a hesitant fuzzy normed space (V,H, ∗) is defined as
the convergence of all cauchy sequences in V toward point in V.

Definition 11. The hesitant fuzzy normed space (V,H, ∗) is considered complete if all of
the cauchy sequences in V converge toward point in V .

Definition 12. The sequence {an}∞n=1 in R is said to be hesitant fuzzy bounded if there
exists S∗ ∈ P [0, 1] such that HR(an, t) ⊃ U∗\S∗, ∀t > 0.

Theorem 1. Let (V,HV , ∗) ,(R,HR, ∗) be two hesitant fuzzy normed linear spaces along
with {v1, v2 . . . vn} be linearly independent set in (V,HV , ∗).Then there is ∅∗ ⊂ S∗

3 ⊂ U∗

such that HV [ β1v1 + · · ·+ βnvn , t ] ⊆ S∗
3 ∗ HR(βj , t) for some 1 ≤ j ≤ n.

Proof. If this isn’t the case, we may discover a sequence {vm} in V where vm =
β1mv1+, . . . ,+β1nvn so that limn→∞HV (vm, t) = U∗.
For every fixed j, we now have a sequence βjm = {βj1 . . . , βjm, . . . , ) represents hesitant
fuzzy bounded, because if the sequence {an}∞n=1 in R is hesitant fuzzy approaches the
limit ′a′ then it is hesitant fuzzy bounded. Considering ∅∗ ⊆ HR(βjm, t) ⊆ U∗, so {βjm}



K. Kavitha, P. Muralikrishna / Eur. J. Pure Appl. Math, 18 (4) (2025), 6541 5 of 14

has a convergent subsequence. For every 1 ≤ j ≤ n., let βj represent the limit of the
subsequence {βjm}. An analogous subsequence of scalars βjm converges for βj for every
1 ≤ j ≤ n., where {vjm} represents the corresponding subsequence of {vm}. Now, put
v =

∑∞
j=1 βjvj then {vm} has a subsequence {vjm} converges to v, since {v1, v2, . . . , vn}

is linearly independen set so v ̸= 0. Now {vjm} → v =⇒ The fuzzy continuity of the
hesitant fuzzy norm is Hv(vjm , v). But ,H(vm, 1) → U∗ by our assumption and {vjm}
is a subsequence of {vm}. Thus Hv(vjm , v) → U∗. Hence H(v, t) = U∗. So, v = 0. This
contradicts v ̸= 0.

Theorem 2. Consider a hesitant fuzzy normed space (V,HV , ∗). W is complete if it is a
finite-dimensional subspace of V .

Proof. Assume that the sequence {vm} is Cauchy in W. Assume dim W = n and
B = {w1, w2, . . . , wn} become any basis to W. After that, every vm is represented in a
distinctive way as vm = γ1mw1+, . . . ,+γnmwn. Given that the sequence {vm} is Cauchy,
for any m, n ≥ N. Now by theorem 1, we possess some ∅∗ ⊂ S∗

3 ⊂ U∗ such that

U∗\S∗
3 ⊂ Hv(vm − vn, t) = Hv

(∑n
j=1(γjm − γjn)wj , t ]

)
⊆ S∗

3 ∩ HR(γjm − γjn, t)

=⇒ HR(γjm − γjn, t) ⊃ [U∗ ∩ S∗
5 ]\S∗

3 .
The following demonstrates that (γjm) = (γj1, γj2, . . . , ) is a Cauchy sequence in R or C,
thus γjm → γj for each 1 ≤ j ≤ n.
Let A =

∑
j=1 γjwj ,clearly A ∈ W. Also now for all m > n,

Hv(vm − v, t) = Hv

(∑n
j=1(γjm − γjn)wj , t

)
⊇ Hv

(
w1,

t
n|γ1m−γ1|

)
∩ Hv

(
w1,

t
n|γ2m−γ2|

)
∩, . . . ,∩ Hv

(
w1,

t
n|γnm−γn|

)
Hv(vm − v, t) ⊇ (U∗\S∗31) ∩ (U∗\S∗32) ∩, . . . , (U∗\S∗3n).
Where Hv

(
wj ,

t
n|γjm−γj |

)
= (U∗\S∗

3j)for some ∅∗ ⊂ (U∗\S∗
3j) ⊂ U∗, j = 1, 2, . . . , n.

Let (U∗\S∗31) ∩ (U∗\S∗32)∩, . . . ,∩ (U∗\S∗3n) ⊃ (U∗\S∗4). So,Hv(vm − v, t) ⊃ (U∗\S∗4) where
S∗4 ∈ P[0, 1],∀m > N. Hence vm → v.

Theorem 3. Fuzzy normed space (V,H, ∗) is compact if and only if each {vn} at V
includes {vnk

} using {vnk
} → v.

4. Finite Dimensional Intuitionistic hesitant Fuzzy Normed Linear
Spaces

The completeness along with compactness features regarding intuitionistic hesitant
fuzzy normed linear spaces with finite dimensions are examined in this section.

Definition 13. Intuitionistic Fuzzy Norm : [18]
V is a linear space throughout the field F. Suppose ∗ constitute a continuous t-norm as well
as ⋄ represent a continuous t-co-norm over V, an intuitionistic fuzzy norm is an object
regarding the following form
{((x11, t11), NI(x11, t11),MI(x11, t11)) : (x11, t11) ∈ V×R+},wherein NI,MI have been fuzzy



K. Kavitha, P. Muralikrishna / Eur. J. Pure Appl. Math, 18 (4) (2025), 6541 6 of 14

sets overV ×R+,Nindicates the degree of membership along with Mindicates the degree of
non-membership (x11, t11) ∈ V × R+ staisfying conditions listed below,

(i) NI(x11, t11) +MI(x11, t11) ≤ 1, ∀(x11, t11) ∈ V × R+.

(ii) NI(x11, t11) > 0.

(iii) NI(x11, t11) = 1 iff x11 = 0.

(iv) NI(cx11, t11) = NI(x11,
t11
|c| ), c ̸= 0, c ∈ F

(v) NI(x11, s11) ∗NI(y11, t11) ≤ NI(x11 + y11, s11 + t11).

(vi) NI(x11, �) is non-decreasing function of R+ and limt11→∞NI(x11, t11) = 1.

(vii) MI(x11, t11) > 0.

(viii) MI(x11, t11) = 0 iff x11 = 0.

(ix) MI(cx11, t11) = MI(x11,
t11
|c| ), c ̸= 0, c ∈ F

(x) MI(x11, s11) ⋄MI(y11, t11) ≥ MI(x11 + y11, s11 + t11).

(xi) MI(x11, �) is non-increasing function of R+ as well as limt11→∞MI(x11, t11) = 0.

According to this definition, the 5−tuple (V,MI, NI, ∗, ⋄) constitutes an intuitionistic fuzzy
normed linear space, whereas (MI, NI) represents an intuitionistic fuzzy norm.

Definition 14. Intuitionistic Hesitant Fuzzy Set : [19]
When applied to X , the functions h and h

′
yield subsets regarding [0, 1], which might be

expressed mathematically E = {(x, h(x), H ′
(x))/x ∈ X}. This consists of intuitioistic

hesitant fuzzy set on X , wherein sets of some values in [0, 1] are represented by h(x), h
′
(x),

The elements x ∈ X that represent the membership along with non-membership degrees of
the set E, Suppose that max(h(x))+min(h

′
(x)) ≤ 1along with min(h(x))+max(h

′
(x)) ≤ 1,

because (h(x), h
′
(x)) is an intuitionistic hesitant fuzzy element.

Definition 15. Intuitionistic Hesitant Fuzzy Norm : [19]
Over the field F, V represents a linear space. Let ∗ represent a continuous t-norm, as well
as ⋄ represent a continuous t-co-norm and an item of the following type is an intuitionistic
hesitant fuzzy norm on V
{HIHF = ((x11, t11),NI(x11, t11),MI(x11, t11)) : (x11, t11) ∈ V×R+},wherein NI ,MI are
fuzzy sets overV × R+,NI indicates the degree of membership along with MI denote the
degree of non-membership (x11, t11) ∈ V × R+ meeting the requirements listed here,

(i) NI(x11, t11) ∪MI(x11, t11) ⊆ U∗, ∀(x11, t11) ∈ V × R+.

(ii) NI(x11, t11) ̸= ∅∗.

(iii) NI(x11, t11) = U∗ iff x11 = 0.



K. Kavitha, P. Muralikrishna / Eur. J. Pure Appl. Math, 18 (4) (2025), 6541 7 of 14

(iv) NI(cx11, t11) = NI

(
x11,

t11
|c|

)
, c ̸= 0, c ∈ F

(v) NI(x11, s11) ∗ NI(y11, t11) ⊆ NI(x11 + y11, s11 + t11).

(vi) NI(x11, �) is non-decreasing function of R+ and limt11→∞NI(x11, t11) = U∗.

(vii) MI(x11, t11) ̸= ∅∗.

(viii) MI(x11, t11) = ∅∗ iff x11 = 0.

(ix) MI(cx11, t11) = MI

(
x11,

t11
|c|

)
, c ̸= 0, c ∈ F

(x) MI(x11, s11) ⋄MI(y11, t11) ⊇ MI(x11 + y11, s11 + t11).

(xi) MI(x11, �) is non-increasing function of R+ as well as limt11→∞MI(x11, t11) = ∅∗.

Definition 16. Intuitionistic Hesitant Fuzzy Normed Linear Space: [19]
Assuming that HIHF represents an Intuitionistic Hesitant Fuzzy Norm over V on F, af-
terwards (V,HIHF ) is an intuitionistic hesitant fuzzy normed linear space or IHFNLS.

Example: [19] Consider the normed linear space (V = R, ∥ � ∥), wherein ∥ x ∥=
|x|, ∀x ∈ R. Describe for all S∗1, S

∗
2 ∈ P[0, 1],S∗1 ∗ S∗2 = S∗1 ∩ S∗2 and S∗1 ⋄ S∗2 = S∗1 ∪ S∗2. Also

define

NI(x1, t1) =


∅∗ if t1 = 0 and ∀ x1 > 0 ∈ V,

U∗ if x1 = 0 and ∀ t1 > 0,

S∗ otherwise S∗ ∈ P[0, 1], Where S∗ is an arbitrary subset of P[0, 1].

and

MI(x1, t1) =


U∗ if t1 = 0 and ∀ x1 > 0 ∈ V,

∅∗ if x1 = 0 and ∀ t1 > 0,

S∗ otherwise S∗ ∈ P[0, 1], Where S∗ is an arbitrary subset of P[0, 1].

Definition 17. Let (V,HIHF ) be an intuitionistic hesitant fuzzy normed linear space. If
dimension of a vector space V is finite then it is called finite dimensional intuitionistic
hesitant fuzzy normed linear space.

Definition 18. [19] If given S∗
1 ̸= ∅∗, t > 0, ∅∗ ⊂ S∗

1 ⊂ U∗,∃ n0 ∈ N so that, NIHF (xn −
x1, t1) ⊃ U∗\S∗

1 and MIHF (xn − x1, t1) ⊂ S∗
1 , ∀ n ≥ n0.

Theorem 4. [19] In an IHFNLS (V,HIHF ), a sequence {xn}n converges to x1 ∈ V if and
only if limn→∞ NIHF (xn − x1, t1) = U∗ along with limn→∞ MIHF (xn − x1, t1) = ∅∗.

Theorem 5. [19] In an IHFNLS (V,HIHF ), a sequence {xn}n has a unique limit if it is
convergent.



K. Kavitha, P. Muralikrishna / Eur. J. Pure Appl. Math, 18 (4) (2025), 6541 8 of 14

Lemma 2. Consider an intuitionistic hesitant fuzzy normed linear space (V,HIHF ), where
{x1, x2, . . . , xn} is a set of vectors in V, that are linearly independent along with the un-
derlying t-norm ∗ along with t-co–norm ⋄ are continuous at (0, 0) and (1, 1) respectively.
Then there exists h1, h2 > 0 and there exists S∗

1 , S
∗
2 ∈ P[0, 1] so that for any collection of

scalars, {δ1, δ2, . . . , δn},

NIHF

{
δ1x1 + δ2x2 + . . . ,+δnxn, h1

n∑
k=1

|δk|

}
⊂ U∗\S∗

1 (1)

MIHF

{
δ1x1 + δ2x2 + . . . ,+δnxn, h2

n∑
k=1

|δk|

}
⊃ S∗

2 (2)

Proof. Let T = |δ1| + |δ2| + · · · + |δn|. If T = 0, then δk = 0, ∀k = 1, 2, . . . n and the
relation, NIHF{δ1x1 + δ2x2 + . . . ,+δnxn, h1

∑n
k=1 |δk|} ⊂ U∗\S∗1 is true for any h > 0,and

S∗ ∈ P[0, 1].
Then, we assume that T > 0. Then (1) is equivalent to

NIHF {ω1x1 + ω2x2 + . . . ,+ωnxn, h1} ⊂ U∗\S∗
1 (3)

for any scalars ω′s with
∑n

k=1 |ωk| = 1 and for some h1 > 0 and S∗ ∈ P[0, 1].
Assume (3) is not true, if at all possible. Consequently, for each h > 0 and S∗

1 ∈ P[0, 1],
there will be a collection regarding scalars {ω1, ω2, . . . ωn} using

∑n
k=1 |ωk| = 1 for which,

NIHF {ω1x1 + ω2x2 + . . . ,+ωnxn, h} ⊇ U∗\S∗. Then for h = { 1
m},m = 1, 2, . . . , there

will be a collection regarding scalars {ωm
1 , ωm

2 , . . . , ωm
n } with

∑n
k=1 |ωm

k | = 1 so that
NIHF(ym, 1

m) ⊃ U∗\{ 1
m} where ym = ωm

1 x1 + ωm
2 x2+, . . . , ωm

n xn. since,
∑n

k=1 |ωm
k | = 1,

we have 0 ≤ |ωm
k | ≤ 1 for k = 1, 2, . . . n. Consequently, ωm

1 has a convergent subsequence
since the sequence {ωm

k } is confined for each fixed k. Let ω1 represent the subsequence’s
limit, as well as allow {y1m} represent the equivalent subsequence regarding {ym}. The
equivalent subsequence of scalars {ωm

2 } converges to ω2. for the subsequence {y1m}, ac-
cording to the same argument. Following this procedure, we get a subsequence after n
steps, {ynm} whereas y1m =

∑n
k=1 η

m
k xk with

∑n
k=1 |ηmk | = 1, and ηmk → ωk as m → ∞.

Let y = η1x1 + · · · + ηkxk Now we show that limm→∞NIHF (yn,m − y, t) = U∗, ∀ t > 0.
We possess NIHF (yn,m − y, t) = NIHF (

∑n
k=1(η

m
k − ωk)xk, t)

⊇ NIHF (x1,
t

n|ηm1 −ω1|) ∗ · · · ∗ NIHF (x1,
t

n|ηmn −ωn|)

So, limm→∞NIHF (yn,m−y, t) ⊇ limm→∞NIHF (x1,
t

n|ηm1 −ω1|)∗· · ·∗limm→∞NIHF (x1,
t

n|ηmn −ωn|)

=⇒ limm→∞NIHF (yn,m−y, t) ⊇ U∗ ∗· · ·∗U∗ (through the t-norm ∗’s continuity at (1,1))

lim
m→∞

NIHF (yn,m − y, t) = U∗, ∀t > 0. (4)

Select m so that 1
m < l. for l > 0. We have

limm→∞NIHF (yn,m, l) = limm→∞NIHF (yn,m+0, 1
m + l− 1

m) ⊇ limm→∞NIHF (yn,m, 1
m)∗

limm→∞NIHF (0, l − 1
m)

⊇ (U∗\ 1
m) ∗ U∗ = U∗\ 1

m =⇒ limm→∞NIHF (yn,m, l) ⊇ U∗

=⇒ lim
m→∞

NIHF (yn,m, l) = U∗ (5)



K. Kavitha, P. Muralikrishna / Eur. J. Pure Appl. Math, 18 (4) (2025), 6541 9 of 14

Now, limm→∞NIHF (y, 2l) = limm→∞NIHF (y − yn,m + yn,m, l + l)
⊇ limm→∞NIHF (y − yn,m, l) ∗ limm→∞NIHF (yn,m, l)
=⇒ limm→∞NIHF (y, 2l) ⊇ U∗ ∗ U∗ (through the t-norm ∗’s continuity at (1,1))
=⇒ limm→∞NIHF (y, 2l) = U∗ ∗U∗ = U∗. (By (4) and (5)) This is because l > 0 becomes
random. In addition, given that

∑n
k=1 |ωm

k | = 1,the linear independence of the vectors
{x1, x2, . . . xn} is established. Consequently, y = ω1x1 + ω2x2 + . . . ,+ωnxn ̸= 0.The
result is a contradiction. We now demonstrate the relationship. MIHF {δ1x1 + δ2x2 +
. . . ,+δnxn, h2

∑n
k=1 |δk|} ⊃ S∗

2 . If T = 0, then δk = 0,∀k = 1, 2, . . . n and the relation,
MIHF {δ1x1 + δ2x2 + . . . ,+δnxn, h2

∑n
k=1 |δk|} ⊃ S∗

2 remains true for every h > 0, and
S∗ ∈ P[0, 1].
Then, assuming that T > 0, (2) is equal to

MIHF {ω1x1 + ω2x2 + . . . ,+ωnxn, h2} ⊃ S∗
2 (6)

for any scalars ω′s with
∑n

k=1 |ωk| = 1. and for some h2 > 0 and S∗
2 ∈ P[0, 1].

If at all feasible, assume that (6) is not true. Consequently, for every h > 0 and S∗ ∈
P[0, 1], There is such a collection regarding scalars {ω1, ω2, . . . ωn}along side

∑n
k=1 |ωk| = 1

that,
MIHF {ω1x1+ω2x2+ . . . ,+ωnxn, h} ⊆ S∗. Then for h = { 1

m},m = 1, 2, . . . , A collection
of scalars {ηm1 , ηm2 , . . . , ηmn } exists. with

∑n
k=1 |ηmk | = 1 so that MIHF (zm, 1

m) ⊆ { 1
m}

where zm = ηm1 x1+, ηm2 x2+, . . . , ηnx
m
n . since,

∑n
k=1 |ηmk | = 1, we have 0 ≤ |ηmk | ≤ 1

regarding k = 1, 2, . . . n. Afterwards, using the similar justification as before, we obtain
a subsequence {znm} where znm =

∑n
k=1 ξ

m
k xk with

∑n
k=1 |ξmk | = 1, and ξmk → ξk as

m → ∞. Thus
∑n

k=1 |ξk| = 1.
Let z = ξ1x1 + · · ·+ ξkxk.Then we have

lim
m→∞

MIHF (zn,m − z, t) = ∅∗, ∀t > 0. (7)

In this case, l > 0, select m so that 1
m < l. We now possess

limm→∞MIHF (zn,m, l) = limm→∞MIHF (zn,m + 0, 1
m + l − 1

m)
⊆ limm→∞MIHF (zn,m, 1

m) limm→∞MIHF (0, l − 1
m) ⊆ ∅∗ ⋄ ∅∗ = ∅∗

=⇒ limm→∞MIHF (zn,m, l) ⊆ ∅∗

=⇒ lim
m→∞

MIHF (zn,m, l) = ∅∗ (8)

Now, limm→∞MIHF (z, 2l) = limm→∞MIHF (z − zn,m + zn,m, l + l)
⊆ limm→∞MIHF (z − zn,m, l) ⋄ limm→∞MIHF (zn,m, l)
=⇒ limm→∞MIHF (z, 2l) ⊆ ∅∗ ⋄ ∅∗ (according to tco–norm ⋄’s continuity at (0,0))
=⇒ limm→∞MIHF (z, 2l) = ∅∗ ⋄ ∅∗ = ∅∗. (By (7) and (8))
Assuming that l > 0 is random. Therefore, z = 0. once again because

∑n
k=1 |ξmk | = 1

along with {x1, x2, . . . xn} is a collection of vectors that are linearly independent. So
z = ξ1x1 + ξ2x2 + . . . ,+ξnxn ̸= 0.Consequently, This leads to a contradiction, This brings
the lemma to an end.



K. Kavitha, P. Muralikrishna / Eur. J. Pure Appl. Math, 18 (4) (2025), 6541 10 of 14

Theorem 6. All finite-dimensional IHFnNLS (V,HIHF ) are complete if the underlying
t-norm ∗ at (1, 1) together with t-co-norm ⋄ at (0, 0) are continuous.

Proof. Assume that dimV = k and that (V,HIHF ) is an intuitionistic hesitant fuzzy
normed linear space. Consider {xn} to become a cauchy sequence in V and {e1, e2, . . . , ek}
providing a basis of V. Let xn = ωn

1 e1 + ωn
2 e2+, . . . , ωn

k ek. where ωn
1 , ω

n
2 , . . . , ω

n
k are

appropriate scalars. Thus,

lim
m,n→∞

NIHF (xm − xn, t) = U∗, ∀ t > 0 (9)

along with
lim

m,n→∞
NIHF (xm − xn, t) = ∅∗, ∀ t > 0 (10)

It is evident out of lemma (2) that ,there exists h1, h2 > 0 and S∗
1 , S

∗
2 ∈ P[0, 1] such that

NIHF

(
k∑

i=1

(ωm
i − ωn

i )ei, h1

k∑
i=1

(|ωm
i − ωn

i |)

)
⊂ U∗\S∗

1 (11)

MIHF

(
k∑

i=1

(ωm
i − ωn

i )ei, h2

k∑
i=1

(|ωm
i − ωn

i |)

)
⊃ S∗

2 (12)

Again for U∗ ⊃ S∗
1 ⊃ ∅∗ out of (9), Consequently, a positive integer n0 exists with regard

to

NIHF

(
k∑

i=1

(ωm
i − ωn

i )ei, t

)
⊃ U∗\S∗

1 , ∀ m,n ≥ n0. (13)

and for U∗ ⊃ S∗
2 ⊃ ∅∗ out of (10), consequently, a positive integer exists. m0 such that

MIHF

(
k∑

i=1

(ωm
i − ωn

i )ei, t

)
⊂ S∗

2 , ∀ m,n ≥ m0. (14)

Now from (11) and (13) we have, NIHF

(∑k
i=1(ω

m
i − ωn

i )ei, t
)
⊃ U∗\S∗

1

⊃ NIHF

(∑k
i=1(ω

m
i − ωn

i )ei, h1
∑k

i=1(|ωm
i − ωn

i |)
)

∀ m,n ≥ n0.

=⇒ h1
∑k

i=1(|ωm
i − ωn

i |) < t,∀ m, n ≥ n0 (Given that NI(x, �) is non decreasing in t)

=⇒
∑k

i=1(|ωm
i − ωn

i |) < t
h1
, ∀ m, n ≥ n0 =⇒ |ωm

i − ωn
i | < t

h1
, ∀ m, n ≥ n0 alon with

i = 1, 2, . . . k. Given that t > 0 is random, based on previously mentioned, we possess
limm,n→∞ |ωm

i − ωn
i | = 0 for i = 1, 2 . . . , k. =⇒ {ωn

i } this constitutes a cauchy se-
quence of scalars for every i = 1, 2 . . . , k.. Thus, every sequence {ωn

i } converges. Let

limn→∞ ωn
i = ωi regarding i = 1, 2 . . . , k. along with x =

∑k
i=1 ωiei. Obviously, x ∈ V.

Afterwards
∀ t > 0, NIHF (xn−x, t) = NIHF

(∑k
i=1 ω

n
i ei −

∑k
i=1 ωiei, t

)
= NIHF

(∑k
i=1(ω

n
i − ωi)ei, t

)
That is

NIHF (xn − x, t) ⊇ NIHF

(
e1,

t

k|ωn
1 − ω1|

)
∗ · · · ∗ NIHF

(
ek,

t

k|ωn
k − ωk|

)
(15)



K. Kavitha, P. Muralikrishna / Eur. J. Pure Appl. Math, 18 (4) (2025), 6541 11 of 14

when n → ∞, then t
k|ωn

i −ωi| → ∞ (Since ωn
i → ωi) for i = 1, 2 . . . , k. and t > 0. Utilizing

the t-norm ∗ continuity at (1,1), we derive from (15)
limn→∞NIHF (xn − x, t) ⊇ U∗ ∗ · · · ∗ U∗, ∀ t > 0.

=⇒ lim
n→∞

NIHF (xn − x, t) = U∗, ∀ t > 0. (16)

Now from (12) and (14) we have, MIHF

(∑k
i=1(ω

m
i − ωn

i )ei, t
)
⊂ S∗

2

⊂ MIHF

(∑k
i=1(ω

m
i − ωn

i )ei, h2
∑k

i=1(|ωm
i − ωn

i |)
)

∀ m, n ≥ n0.

=⇒ h2
∑k

i=1(|ωm
i − ωn

i |) < t, ∀ m, n ≥ n0 (Because MI(x, �) has non increasing in t)

=⇒
∑k

i=1(|ωm
i − ωn

i |) < t
h2
, ∀ m, n ≥ m0 =⇒ |ωm

i − ωn
i | < t

h2
, ∀ m,n ≥ m0 in addition

i = 1, 2, . . . k. Given that t > 0 is random, we can derive limm,n→∞ |ωm
i − ωn

i | = 0 over
i = 1, 2 . . . , k. for every i = 1, 2 . . . , k. is a cauchy sequence of scalars. Thus, every sequence
{ωn

i } converges. Let limn→∞ ωn
i = ωi for i = 1, 2 . . . , k. together with x =

∑k
i=1 ωiei.

Evidently x ∈ V. Afterwards ∀ t > 0,

MIHF (xn − x, t) = MIHF

(∑k
i=1 ω

n
i ei −

∑k
i=1 ωiei, t

)
= MIHF

(∑k
i=1(ω

n
i − ωi)ei, t

)
That is

MIHF (xn − x, t) ⊇ MIHF

(
e1,

t

k|ωn
1 − ω1|

)
⋄ · · · ⋄MIHF

(
ek,

t

k|ωn
k − ωk|

)
(17)

when n → ∞, then t
k|ωn

i −ωi| → ∞ (Since ωn
i → ωi) for i = 1, 2 . . . , k. and t > 0. Applying

the t-co-norm’s continuity ⋄ at (0,0), we obtain from (17).
limn→∞MIHF (xn − x, t) ⊇ ∅∗ ⋄ · · · ⋄ ∅∗, ∀ t > 0.

=⇒ lim
n→∞

MIHF (xn − x, t) = ∅∗, ∀ t > 0. (18)

We obtain xn → x as n → ∞. by (16) and (18). Consequently (V,HIHF ) is complete.

Definition 19. Consider the IHFNLS (V,HIHF ) and X ⊂ V. For every S∗, X is consid-
ered to be bounded, ∅∗ ⊂ S∗ ⊂ U∗, ∃t1, t2 > 0 such that NIHF (x, t1) ⊃ U∗\S∗ and
MIHF (x, t2) ⊂ S∗, ∀x ∈ X.

Theorem 7. A subset X is compact if and only if it is closed and bounded in the finite
dimensional IHFNLS (V,HIHF ), where both the t-co-norm ⋄ along with the underlying
t-norm ∗ are continuous at (0,0) together with (1,1) respectively.

Proof. We start by assuming that X is compact. We need for prove it X is bounded
alongwith closed. Take x ∈ X. Then limn→∞ xn = x. indicates that there is a sequence
{xn} in X. There is a subsequence {xnk

} of {xn} that converges to a point in X because
X is compact. Afterwards, {xn} → x, and since x ∈ X, X is closed. If at all possible,
assume that X is unbounded. Then there exists a S∗

0 with ∅∗ ⊂ S∗
0 ⊂ U∗ so that for

any positive number n, there exists x0 ∈ X in a manner that,NIHF (xn, n) ⊆ U∗\S∗
0 or



K. Kavitha, P. Muralikrishna / Eur. J. Pure Appl. Math, 18 (4) (2025), 6541 12 of 14

MIHF (xn, n) ⊇ S∗
0 . So there exists a subsequence of {xn} whereby minimum of one of the

relationships have
NIHF (xnk

, nk) ⊆ U∗\S∗
0 , ∀ n ∈ N. (19)

MIHF (xnk
, nk) ⊇ S∗

0 , ∀ n ∈ N. (20)

possesses. Initially, we consider that NIHF (xnk
, nk) ⊆ U∗\S∗

0 , ∀n ∈ N holds. Now for
t > 0,
U∗\S∗

0 ⊇ NIHF (xnk , nk) = NIHF (xnk − x+ x, nk − t + t) where t > 0.
=⇒ U∗\S∗

0 ⊇ NIHF (xnk , nk) = NIHF (xnk − x, t) ∗ NIHF (x, nk − t)
=⇒ U∗\S∗

0 ⊇ limk→∞NIHF (xnk − x, t) ∗ limk→∞NIHF (x, nk − t)
=⇒ U∗\S∗

0 ⊇ U∗∗U∗ = U∗. (Applying the t-norm’s continuity at (1,1) ) =⇒ U∗\S∗
0 ⊇ U∗

which is a contradiction. In case MIHF (xnk
, nk) ⊇ S∗

0 , ∀n ∈ N possess , examining the
function MIHF (x, t) and continuing as previously mentioned, We get a contradiction,
Hence X is bounded.
Conversly, Assuming that X is bounded and closed, we must demonstrate that X is
compact. Consider dimV = n along with {e1, e2, . . . en} as a basis of V respectively. Select

{xk} as a sequence inXwhile assume xk = ω
(k)
1 e1+· · ·+ω

(k)
n en here ω

(k)
1 , . . . ω

(k)
n are scalars.

Now, according to lemma (1), there is h1, h2 > 0 and there exists S∗
1 , S

∗
2 ∈ P[0, 1] such

that

NIHF

(
n∑

i=1

ω
(k)
i ei, h1

n∑
i=1

|ω(k)
i |

)
⊂ U∗\S∗

1 (21)

and

MIHF

(
n∑

i=1

ω
(k)
i ei, h2

n∑
i=1

|ω(k)
i |

)
⊃ S∗

2 (22)

Again since X is bounded, for S∗
1 ∈ P[0, 1],∃ t1 > 0, so that

NIHF (x, t1) ⊃ U∗\S∗
1 and there exists t2 > 0, so that MIHF (x, t2) ⊂ S∗

1 , ∀ x ∈ X. So,

NIHF

(
n∑

i=1

ω
(k)
i ei, t1

)
⊃ U∗\S∗

1 (23)

and

MIHF

(
n∑

i=1

ω
(k)
i ei, t1

)
⊂ S∗

2 (24)

from (21) and (23) we get ,

NIHF

(∑n
i=1 ω

(k)
i ei, h1

∑n
i=1 |ω

(k)
i |
)
⊂ U∗\S∗

1 ⊂ NIHF

(∑n
i=1 ω

(k)
i ei, t1

)
=⇒ NIHF

(∑n
i=1 ω

(k)
i ei, h1

∑n
i=1 |ω

(k)
i |
)
⊂ NIHF

(∑n
i=1 ω

(k)
i ei, t1

)
=⇒ h1

∑n
i=1 |ω

(k)
i | < t1 (Given that NIHF (x, �) is non- decreasing)

=⇒ |ω(k)
i | ≤ t1

h1
where i = 1, 2 . . . n. and k = 1, 2 . . . Thus, for every {ω(k)

i }It is bounded
(i = 1, 2, . . . n). As a result of repeatedly applying the Bolzano -Weierstrass theorem, every

sequence in {ω(k)
i } possesses a subsequence that converges {ω(kl)

i }, ∀ i = 1, 2, . . . n. Assume



K. Kavitha, P. Muralikrishna / Eur. J. Pure Appl. Math, 18 (4) (2025), 6541 13 of 14

that xkl = ω
(kl)
i e1 + · · · + ω

(kl)
i en and that {ω(kl)

1 }, {ω(kl)
2 }, . . . {ω(kl)

n } are all convergent.

Consider ωi = liml→∞ ω
(kl)
i , i = 1, 2 . . . n. and x = ω1e1 + ω2e21 + · · · + ωnen. Currently,

for t > 0,

we possess NIHF (xkll − x, t) = NIHF

(∑n
i=1(ω

(kl)
i − ωi)ei, t)

)
⊇ NIHF

(
e1,

t

|ω(kl)
1 −ω1|

)
∗ · · · ∗ NIHF

(
en,

t

|ω(kl)
n −ωn|

)
=⇒ liml→∞NIHF (xkl − x, t) ⊇ U∗ ∗ · · · ∗ U∗.(ω

(kl)
n → ωi as l → ∞)(With the use of

contuity of t- norm ∗ at (1, 1)).

=⇒ lim
l→∞

NIHF (xkl − x, t) = U∗ (25)

Currently, for t > 0, we possess

MIHF (xkll − x, t) = MIHF

(∑n
i=1(ω

(kl)
i − ωi)ei, t)

)
⊆ MIHF

(
e1,

t

|ω(kl)
1 −ω1|

)
⋄ · · · ⋄MIHF

(
en,

t

|ω(kl)
n −ωn|

)
=⇒ liml→∞MIHF (xkl − x, t) ⊇ ∅∗ ⋄ · · · ⋄ ∅∗.(ω(kl)

n → ωiasl → ∞)(using the contuity of
t-co- norm ⋄ at (0,0)).

=⇒ lim
l→∞

NIHF (xkl − x, t) = ∅∗ (26)

According to (25) and (26), x(kl) → x. Given that X is closed, x ∈ X implies that X is
compact. Thus, the proof is completed.

5. Conclusion

We extended the foundational work on fuzzy normed linear space, initially proposed by
Samanta et al., to the domain of hesitant fuzzy normed linear space and also examine the
properties of completeness and compactness on hesitant fuzzy normed linear space, also
address the same properties on intuitionistic hesitant fuzzy normed linear space in finite
dimension using definitions, lemmas, and theorems. We also investigate the continuity
of underlying t-norms and co-t-norm on finite-dimensional intuitionistic hesitant fuzzy
normed linear space. In this context, there is room for more work (If possible in practical
scenario).

Acknowledgements

The authors are grateful to Dr. M. Padmini, Head, Department of Mathematics, who
motivated with the right guidance and support.

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