EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 2, Article Number 5565
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Bounded q-variation Functions in Spaces with Indefinite
Metric

Osmin Ferrer Villar1,∗, José Naranjo Mart́ınez2

1 Departament of Mathematics, University of Sucre, Cra. 28 # 5-267, Puerta Roja,
Sincelejo, Sucre, Colombia
2 Center for Basic Sciences, School of Engineering and Architecture, Pontifical Bolivarian
University, Monteria, Colombia

Abstract. In this paper we establish the definition of bounded q-variation functions on Krein
spaces (definition 7) and give a complete characterisation by comparing it with bounded q-variation
functions on classical Hilbert spaces and with Hilbert spaces associated to a Krein space (theorem
6). The fundamental tools of the theory of bounded q-variation functions in the formalism of Krein
spaces are described (theorems 7, 8). Furthermore, we endow the spaces of bounded q-variation
functions in Krein spaces with a norm, which we have called q-norm (theorem 13).

2020 Mathematics Subject Classifications: 46C20, 26A45, 46B99, 47B50

Key Words and Phrases: Krein space, fundamental symmetry, q-variation, total q-variation,
bounded q-variation, q-norm.

1. Introduction

In 1881, Jordan discovered in the work of Dirichlet the notion of function of bounded
variation and proves that for this class of functions the Fourier conjecture is valid. He
also shows that the function f : [a, b] → R has bounded variation on [a, b] if and only
if f is the difference of monotone functions (nowadays this result is known as Jordan’s
Representation Theorem). The notion of function of bounded variation has been studied
and generalized in different contexts, studying different structures and properties in spaces
with research interest. For example, Chistyakov in [1–3] studied a concept of a function of
bounded generalized variation in the sense of Jordan-Riesz-Orlicz for functions f : [a, b]→
X, where X is a normed or metric space. More recently, the notion of functions of
bounded variation in spaces with indefinite metric was introduced by Ferrer, Guzmán and
Naranjo in [4, 5]. Furthermore, functions of bounded variation have multiple applications
in various fields, e.g. in image processing, Brokman, Burger and Gilboa in [6] presented

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5565

Email addresses: osmin.ferrer@unisucre.edu.co (O. Ferrer), jose.naranjom@upb.edu.co (J. Naranjo)

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



O. Ferrer, J. Naranjo / Eur. J. Pure Appl. Math, 18 (2) (2025), 5565 2 of 18

an analysis of the total-variation (TV) on non-Euclidean parametrized surfaces, a natural
representation of the shapes used in 3D graphics, see [7]. Among the results achieved in
this research is a new way to generalize the convexity of sets from the plane to surfaces is
derived by characterizing the TV eigenfunctions on surfaces. Additionally, Bugajewska,
Bugajewski and Hudzik in [8] they investiged solutions of nonlinear Hammerstein and
Volterra−Hammerstein integral equations in the space of functions of bounded ϕ-variation
in the sense of Young. Later, Bugajewska, Bugajewski and Lewicki in [9] explored with the
superposition operator as well as with solutions to non-linear integral equations in spaces
of functions of generalized bounded ϕ-variation. In the field of nonlinear analysis, Xie,
Liu, Li and Huang in [10] they examine the bounded variation capacity (BV capacity) and
characterise the Sobolev-type inequalities associated with BV functions within a general
framework of strictly local Dirichlet spaces with a doubling measure, utilising the BV
capacity.

2. Preliminaries

The following is a type of spaces with indefinite metric, called Krein spaces, which are
a generalisation of Hilbert spaces.

Definition 1. [11, 12] A space K with an indefinite inner product [·, ·] which admits a
fundamental decomposition of the form K = K+ ˙[+]K− such that (K+, [·, ·]) and (K−,−[·, ·])
are Hilbert spaces, is called a Krein space.

Definition 2. Let (K, [·, ·]) be a Krein space with fundamental decomposition K = K+ ˙[+]K−,
then we define the operator J : (K, [·, ·]) −→ (K, [·, ·])

J k = k+ − k−,

it is called the fundamental symmetry of the Krein space K associated with the fun-
damental decomposition. From now on we will write (K, [·, ·],J ) to denote the Krein
space with fundamental symmetry J associated with the fundamental decomposition K =
K+ ˙[+]K−.

Remark 1. Let (K = K+
˙[+]K−, [·, ·],J ) be a Krein space, and f : [a, b]→ K = K+

˙[+]K−.
Considering that for any t in [a, b], f(t) belongs to K = K+

˙[+]K−, we will henceforth write
the image of t under f as f(t) = f+(t) + f−(t), for any t in [a, b].

Remark 2. Let (K = K+
˙[+]K−, [·, ·],J ) be a Krein space, and f : [a, b]→ K = K+

˙[+]K−.
Considering that for any t in [a, b], f(t) ∈ K, therefore there exist k+ ∈ K+ and k− ∈ K−
such that f(t) = k+ + k−, we will write k+ = f+(t) and k− = f−(t). Therefore,

(J f)(t) = J (f(t)) = J (k+ + k−) = k+ − k− = f+(t)− f−(t).

Definition 3. [11] Let (K = K+ ˙[+]K−, [·, ·]) be a Krein space and J the fundamental
symmetry associated to the given decomposition. We define the function
[·, ·]J : K ×K −→ C by

[x, y]J = [J x, y], for all x, y in K.



O. Ferrer, J. Naranjo / Eur. J. Pure Appl. Math, 18 (2) (2025), 5565 3 of 18

This function is a usual inner product and is called J -inner product.

Definition 4. [11, 12] The fundamental symmetry J associated with the Krein space
(K = K+ ˙[+]K−, [·, ·]) induces a norm in K defined by:

∥x∥J :=
√
[x, x]J , for all x in K,

this norm is called the J -norm of K. Explicitly,

∥x∥J = ([x+, x+]− [x−, x−])1/2, for all x in K.

Remark 3. It defines for x+ in K+ and x− in K−

∥x+∥+ =
√
[x+, x+], and ∥x−∥− =

√
−[x−, x−]

From now on, the topology studied in Krein spaces is directly related to the J -norm
of K.

Theorem 1. [4] Let (K = K+
˙[+]K−, [·, ·],J ) be a Krein space, then

∥x∥J ≤ ∥x+∥+ + ∥x−∥− for all x = x+ + x− in K.

Theorem 2. [11, 12] Let (K, [·, ·]) be a Krein space and let

K = K+
1

˙[+]K−
1 , K = K+

2
˙[+]K−

2 ,

be two fundamental decompositions. If J1 and J2 are the respective fundamental symme-
tries it follows that ∥.∥J1 and ∥.∥J2 are equivalent norms.

Example 1. Let us consider the vector space C2, with sum, usual product and the mapping
[·, ·] : C2 × C2 −→ C given by:

[(x1, y1), (x2, y2)] = x1x2 − y1y2. (1)

The space (C2, [·, ·]) is a Krein space with fundamental decomposition C2 = K+[∔]K−,
where K+ = {(x, 0) : x ∈ C} and K− = {(0, y) : y ∈ C}, with fundamental symmetry

J ((x, y)) = J ((x, 0) + (0, y)) = (x,−y) that determines the J -norm ∥ · ∥J given by

∥(x, y)∥J = [J (x, y), (x, y)]1/2 = (x · x− (−y) · y)1/2 =
√
|x|2 + |y|2.

In addition, the norms ∥ · ∥+ : K+ → R+ ∪ {0} and ∥ · ∥− : K− → R+ ∪ {0} are given by:

∥(x, 0)∥+ =
√

[(x, 0), (x, 0)] =
√
|x|2 = |x|

∥(0, y)∥− =
√
[(0, y), (0, y)] =

√
−y(−y) =

√
|y|2 = |y|



O. Ferrer, J. Naranjo / Eur. J. Pure Appl. Math, 18 (2) (2025), 5565 4 of 18

3. Functions of bounded variation in Krein spaces

This section presents the fundamental theory of bounded variation functions in Krein
spaces, which were introduced by Ferrer, Naranjo and Guzmán [4].

Definition 5. [4] Let (K = K+
˙[+]K−, [·, ·]) be a Krein space and let f : [a, b]→ K defined

in [a, b], we will say that f is strongly of bounded variation in [a, b] on (K = K+
˙[+]K−) if

Vb
a(f, (K, [·, ·])) = sup

P∈P[a,b]

{
n∑

i=1

(∥f+(ti)− f+(ti−1)∥+ + ∥f−(ti)− f−(ti−1)∥−)

}
is finite. The set of all functions strongly of bounded variation in [a, b] on
(K = K+

˙[+]K−, [·, ·]) is denoted as follows BV ([a, b],K, [·, ·])

Definition 6. [4] (Positive and negative variations of functions in Krein spaces)
Let (K = K+

˙[+]K−, [·, ·],J ) be a Krein space, and f : [a, b] → K = K+
˙[+]K−, with

f(t) = f+(t) + f−(t), for all t in the interval [a, b]. The positive and negative variation of
f on [a, b] with respect to (K+, [·, ·]) and (K−,−[·, ·]) respectively, are defined by:

+

Vb
a(f, (K+, [·, ·])) = sup

P∈P[a,b]

{
n∑

i=1

∥f+(ti)− f+(ti−1)∥+

}
and

−

Vb
a(f, (K−,−[·, ·])) = sup

P∈P[a,b]

{
n∑

i=1

∥f−(ti)− f−(ti−1)∥−

}
.

Theorem 3. [4] Let (K = K+
˙[+]K−, [·, ·],J ) be a Krein space, f+ : [a, b]→ K+ and

f− : [a, b]→ K− strongly of bounded variation in the Hilbert spaces (K+, [·, ·]) and
(K−,−[·, ·]) respectively, then f : [a, b] → K defined as follows f(t) = f+(t) + f−(t) is
strongly of bounded variation in the Hilbert space (K, [·, ·]J ).

Theorem 4. [4] Let (K = K+
˙[+]K−, [·, ·]) be a Krein space, if f : [a, b] → K is strongly

of bounded variation in (K = K+
˙[+]K−, [·, ·]), then J f is strongly of bounded variation in

(K = K+
˙[+]K−, [·, ·]) and (K, [·, ·]J ).

Theorem 5. [4] Let (K = K+
˙[+]K−, [·, ·]) be a Krein space and let f : [a, b]→ K a strongly

of bounded variation function in [a, b] on (K = K+
˙[+]K−), suppose that c in (a, b), then f

is strongly of bounded variation in [a, c] and in [c, b] on (K = K+
˙[+]K−). In this case we

have
V b
a (f, (K, [·, ·])) = V c

a (f,K) + V b
c (f,K)

4. Functions of bounded q-variation in Hilbert spaces associated with a
Krein space

Given a Krein space (K = K+
˙[+]K−, [·, ·],J ) and f : [a, b] → K = K+

˙[+]K−, for t in
[a, b], as f(t) in K, we can write f(t) = f+(t) + f−(t), where f+(t) in K+ and f−(t) in



O. Ferrer, J. Naranjo / Eur. J. Pure Appl. Math, 18 (2) (2025), 5565 5 of 18

K−. Now, since (K+, [·, ·]) and (K−,−[·, ·]) are Hilbert spaces, the q−variations of f on

[a, b] denoted by

q+

Vb
a(f, (K+, [·, ·])) and

q−

Vb
a(f, (K−, [·, ·])) are given as follows:

q+

Vb
a(f, (K+, [·, ·])) = sup

P∈P[a,b]


(

n∑
i=1

∥f+(ti)− f+(ti−1)∥q+

)1/q


and
q−

Vb
a(f, (K−,−[·, ·])) = sup

P∈P[a,b]


(

n∑
i=1

∥f−(ti)− f−(ti−1)∥q−

)1/q
 .

Where the supreme is taken over the entire set of partitions {t0, t1, t2, . . . , tn} of the interval
[a, b]. We will call these q−variations positive and negative q−variations respectively.

Remark 4. Note that

(i)

q+

Vb
a(f, (K+, [·, ·])) ≥ 0,

q−

Vb
a(f, (K−,−[·, ·])) ≥ 0 and

q

Vb
a(f, (K, [·, ·])) ≥ 0.

(ii) Since (K+, [·, ·]) and (K−,−[·, ·]) are Hilbert space. Given α ∈ C and f, g functions
functions of bounded q-variation, for [13] it is satisfied that:

i)

q+

Vb
a(αf, (K+, [·, ·])) = |α|

q+

Vb
a(f, (K+, [·, ·]))

ii)

q+

Vb
a((f + g), (K+, [·, ·])) =

q+

Vb
a(f, (K+, [·, ·])) +

q+

Vb
a(g, (K+, [·, ·]))

iii)

q−

Vb
a(αf, (K−, [·, ·])) = |α|

q−

Vb
a(f, (K−, [·, ·]))

iv)

q−

Vb
a((f + g), (K−, [·, ·])) =

q−

Vb
a(f, (K−, [·, ·])) +

q−

Vb
a(g, (K−, [·, ·]))

Example 2. Consider example 1 and the function f : [2, 3]→ C2 defined by
f(t) = (ti,−ti) = (ti, 0) + (0,−ti). Aśı, f+(t) = (ti, 0), f+(t) = (ti, 0) and
f−(t) = (0,−ti). Then the positive and negative q−variations are given by:

q+

Vb
a(f, (K+, [·, ·])) = sup

P∈P[a,b]


 n∑

j=1

(tj − tj−1)
q

1/q
 =

q−

Vb
a(f, (K−, [·, ·]))

The existence of functions of bounded q−variation on Hilbert spaces associated to a
Krein space led us to think about the definition of functions of bounded q−variation on
Krein spaces studied in the next section.



O. Ferrer, J. Naranjo / Eur. J. Pure Appl. Math, 18 (2) (2025), 5565 6 of 18

5. Functions of bounded q-variation in Krein spaces

In this section we will introduce in Krein spaces the concepts of total q-variation and
give the notion of a bounded q-variation function, notions that generalise the results given
in [4] and [13]

Definition 7. Let (K = K+
˙[+]K−, [·, ·]) be a Krein space, f : [a, b]→ K and q ≥ 1.at the

number

q

Vb
a(f, (K, [·, ·])) = sup

P∈P[a,b]


(

n∑
i=1

(∥f+(ti)− f+(ti−1)∥+ + ∥f−(ti)− f−(ti−1)∥−)q
)1/q


where the supremum is taken over the set of all partitions P = {t0, t1, . . . , tn} of the
interval [a, b], we will call total q-variation of f in [a, b] on K.

Moreover, if there is a constant M > 0 such that

q

Vb
a(f, (K, [·, ·])) ≤M we will say that f

is of bounded q-variation in [a, b] on K.

Remark 5. The set of all functions of bounded q-variation in [a, b] on (K = K+
˙[+]K−, [·, ·])

will be denoted as follows B
q

V b
a (K, [·, ·]).

(i) Note that when q = 1 we are dealing with definition 4.3 given in [4].

(ii) In the case of a Hilbert space, we have that ∥f−(ti)− f−(ti−1)∥− = 0 and so
q

V b
a (f,K, [·, ·]) = sup

P∈P[a,b]

{(∑n
i=1 ∥f+(ti)− f+(ti−1)∥q+

)1/q}
= Vq(f) which is the

definition of a function of q-variation on a Hilbert space in the Weiner sense [13].

(iii) If f ∈ B
q

V b
a (K, [·, ·]), then f ∈ B

q

V b
a (K+, [·, ·]) and f ∈ B

q

V b
a (K−,−[·, ·]), since that

∥f+(ti) − f+(ti−1)∥+, ∥f−(ti) − f−(ti−1)∥− ≤ ∥f+(ti) − f+(ti−1)∥+ + ∥f−(ti) −
f−(ti−1)∥−

The observation3 is of utmost relevance, since it guarantees that every bounded
q-variation function on Krein space K = K+

˙[+]K− is of bounded q-variation on the asso-
ciated Hilbert spaces K+ and K−.

Remark 6. Note that for the associated Hilbert space (K, [·, ·]J ) the total q-variation of
f is:

q

Vb
a(f, (K, [·, ·]J )) = sup

P∈P[a,b]


(

n∑
i=1

∥f(ti)− f(ti−1)∥qJ

)1/q


Next, we show an example of a bounded q-variation function in a Krein space.



O. Ferrer, J. Naranjo / Eur. J. Pure Appl. Math, 18 (2) (2025), 5565 7 of 18

Example 3. Consider example 1 and the function f : [2, 3]→ C2 defined by
f(t) = (ti,−ti). Let us see that f is a function of bounded 2-variation.
In fact, let P = {t0, t1, . . . , tn} be a partition of [2, 3]. Then,

∥f+(tj)− f+(tj−1)∥+ = ∥(tji− tj−1i, 0)∥+ = |tji− tj−1i| = |tj − tj−1| · |i| = tj − tj−1

∥f−(ji)− f−(tj−1)∥− = ∥(0, tj−1i− tji)∥− = |tj−1i− tji| = |tj−1 − tj | · |i| = tj − tj−1.

Thus,

(∥f+(tj)− f+(tj−1)∥+ + ∥f−(tj)− f−(tj−1)∥−)2 = (2(tj − tj−1))
2 = 4(tj − tj−1)

2

Now, since t ∈ [2, 3], then 0 ≤ tj − tj−1 ≤ 1. Then, (tj − tj−1)
2 ≤ tj − tj−1, so

0 ≤
n∑

j=1

4(tj − tj−1)
2 ≤

n∑
j=1

4(tj − tj−1), it follows that

 n∑
j=1

4(tj − tj−1)
2

1/2

≤

 n∑
j=1

4(tj − tj−1)

1/2

≤ 2 (tn − t0)
1/2 = 2(3− 2)1/2 = 2

Therefore, we have to

2

V3
2(f, (C2, [·, ·])) is bounded, thus it follows that f is of bounded

2-variation function.

In definition 7 we introduce the concept of a bounded q-variation function in Krein
spaces. In the following, we show that this concept is broader than the one given in [13].

Theorem 6. Let (K = K+
˙[+]K−, [·, ·]) be a Krein space and f : [a, b] → K of bounded

q-variation function in (K = K+
˙[+]K−, [·, ·]), then f is of bounded q-variation function in

the Hilbert space (K, [·, ·]J ).

Proof. If f is of bounded q-variation in [a, b] on (K = K+
˙[+]K−, [·, ·]), then exists

M > 0 such that:

q

Vb
a(f, (K, [·, ·])) = sup

P∈P[a,b]

{( n∑
i=1

(∥f+(ti)−f+(ti−1)∥++∥f−(ti)−f−(ti−1)∥−)q
)1/q} ≤M

By Theorem 1 we have that:

∥f(ti)− f(ti−1)∥J ≤ ∥f+(ti)− f+(ti−1)∥+ + ∥f−(ti)− f−(ti−1)∥−

Therefore, for q ≥ 1, it is satisfied that :

∥f(ti)− f(ti−1)∥qJ ≤ (∥f+(ti)− f+(ti−1)∥+ + ∥f−(ti)− f−(ti−1)∥−)q

whence:
q

Vb
a(f, (K, [·, ·]J )) ≤M



O. Ferrer, J. Naranjo / Eur. J. Pure Appl. Math, 18 (2) (2025), 5565 8 of 18

Therefore, f is of bounded q-variation in [a, b] on the Hilbert space (K, [·, ·]J ).

The following result is very significant in this research, as it shows the robustness of
the definition 7 introduced in this paper, more precisely it shows that the bounded q-
variation functions on a Krein space are independent of the fundamental decomposition
of the space. This finding is remarkable as it reinforces the robustness of these functions
in different decomposition structures, this not only challenges traditional conceptions, but
also opens new perspectives in functional analysis, providing a solid basis for future re-
search and applications in economics, quantum mechanics, signal processing[14–17] where
the bounded q-variation plays a very important role.

Theorem 7. Let (K, [·, ·]) be a Krein space with decompositions (K = K1+
˙[+]K1− ,J1),

(K = K2+
˙[+]K2− ,J2) and f : [a, b] → K of bounded q-variation function in [a, b] on

(K = K1+
˙[+]K1−), then f is of bounded q-variation in [a, b] on (K = K2+

˙[+]K2−).

Proof. If f is of bounded q-variation in [a, b] on (K = K1+
˙[+]K1− , [·, ·]), then exists

M > 0 such that:

q

Vb
a(f, (K, [·, ·])) = sup

P∈P[a,b]

{( n∑
i=1

(∥f+(ti)−f+(ti−1)∥1++∥f−(ti)−f−(ti−1)∥1−)q
)1/q} ≤M

Moreover, as ∥ · ∥J1 and∥ · ∥J2 are equivalent norms [12], then there are a, b > 0 such that
a∥ · ∥J2 ≤ ∥ · ∥J1 ≤ b∥ · ∥J2 . Therefore,

(a∥f(ti)− f(ti−1)∥J2)
q ≤ ∥f(ti)− f(ti−1)∥qJ1

≤ (∥f+(ti)− f+(ti−1)∥1+ + ∥f−(ti)− f−(ti−1)∥1−)q

The last inequality is obtained thanks to theorem 1. Later,∑n
i=1(a∥f(ti)− f(ti−1)∥J2)

q ≤
∑n

i=1(∥f+(ti)− f+(ti−1)∥1+ + ∥f−(ti)− f−(ti−1)∥1−)q

(aq
∑n

i=1(∥f(ti)− f(ti−1)∥J2)
q)1/q ≤

(
∑n

i=1(∥f+(ti)− f+(ti−1)∥1+ + ∥f−(ti)− f−(ti−1)∥1−)q)1/q

a(
∑n

i=1(∥f(ti)− f(ti−1)∥J2)
q)1/q ≤

(
∑n

i=1(∥f+(ti)− f+(ti−1)∥1+ + ∥f−(ti)− f−(ti−1)∥1−)q)1/q ≤M
It follows that:

q

Vb
a(f, (K, [·, ·]J2)) = sup

P∈P[a,b]
{(

n∑
i=1

(∥f+(ti)− f+(ti−1)∥2+ + ∥f−(ti)− f−(ti−1)∥2−)q)1/q}

≤M/a

This implies thatf is of bounded q-variation in [a, b] on (K = K2+
˙[+]K2− , [·, ·]J2).



O. Ferrer, J. Naranjo / Eur. J. Pure Appl. Math, 18 (2) (2025), 5565 9 of 18

Theorem 8. Let (K = K+
˙[+]K−, [·, ·]) be a Krein space and f : [a, b] → K of bounded

q-variation function in [a, b] on (K = K+
˙[+]K−, [·, ·]), then f is bounded.

Proof. Suppose that f is of bounded q-variation on [a, b], then there exists M > 0 such

that

q

Vb
a(f, (K, [·, ·]J )) ≤ M for all partition P of [a, b]. Let be t ∈ (a, b) and consider the

partition P = {a, t, b}, then,(
2∑

i=1

(∥f+(ti)− f+(ti−1)∥+ + ∥f−(ti)− f−(ti−1)∥−)q
)1/q

=

((
∥f+(t)− f+(a)∥+ + ∥f−(t)− f−(a)∥−

)q
+
(
∥f+(b)− f+(t)∥+ + ∥f−(b)− f−(t)∥−

)q)1/q
≤ sup

P∈P[a,b]


(

n∑
i=1

(∥f+(ti)− f+(ti−1)∥+ + ∥f−(ti)− f−(ti−1)∥−)q
)1/q

 ≤M

Thus,

((∥f+(t)−f+(a)∥++∥f−(t)−f−(a)∥−)q+(∥f+(b)−f+(t)∥++∥f−(b)−f−(t)∥−)q)1/q ≤M

Therefore,

(∥f+(t)−f+(a)∥++∥f−(t)−f−(a)∥−)q+
(
∥f+(b)− f+(t)∥+ + ∥f−(b)− f−(t)∥−

)q ≤ M q

Whence,

(∥f+(t)− f+(a)∥++ ∥f−(t)− f−(a)∥−)q , (∥f+(b)− f+(t)∥++ ∥f−(b)− f−(t)∥−)q ≤M q

Since (∥f+(t)− f+(a)∥+ + ∥f−(t)− f−(a)∥−)q ≤ M q, then,

∥f+(t)− f+(a)∥+ + ∥f−(t)− f−(a)∥− ≤ M

Then, using the theorem 1, it follows that:

∥f(t)− f(a)∥J ≤ ∥f+(t)− f+(a)∥+ + ∥f−(t)− f−(a)∥− ≤ M

Since, ∥f(t)∥J − ∥f(a)∥J ≤ ∥f(t)− f(a)∥J ,then

∥f(t)∥J ≤ M + ∥f(a)∥J ≤ M + ∥f(a)∥J + ∥f(b)∥J = d.

Therefore for all t ∈ [a, b], it follows that ∥f(t)∥J ≤ d. This implies that f is bounded
in [a, b].

We show below that the reciprocal of 8 is not true.



O. Ferrer, J. Naranjo / Eur. J. Pure Appl. Math, 18 (2) (2025), 5565 10 of 18

Example 4. Consider the Krein space given in example 1 and the function
f : [
√
3, 4]→ C2 defined by

f(t) =


(i, i), if t is rational, t ∈ [

√
3, 4],

(0, 0), if t is irrational, t ∈ [
√
3, 4].

Let’s see that f is bounded on [
√
3, 4]. In fact,

∥f(t)∥J =


√
2, if t is rational, t ∈ [

√
3, 4],

0, if t is irrational, t ∈ [
√
3, 4].

Therefore, ∥f(t)∥J ≤
√
2 for all t ∈ [

√
3, 4]. Thus f is bounded.

Now, let’s see that f is not of q−bounded variation. Let t0 =
√
3, as between any two

reals there is a rational number and an irrational number, we can choose t1 as a rational
number between

√
3 and 4, t2 as an irrational number between t1 and 4, t3 as a rational

number between t2 and 4, and so on t2i would be an irrational number between t2i−1 and
4, t2i+1 would be a rational number between t2i and 4, finally we choose tn = 4. Then,

f+(t) = (i, 0) and f−(t) = (0, i)

Furthermore,

2

V4√
3(f, (C2, [·, ·])) ≥

 n∑
j=1

(
∥f+(tj)− f+(tj−1)∥+ + ∥f−(tj)− f−(tj−1)∥−

)21/2

=
(
(∥ − (i, i)∥+ + ∥(i, i)∥−)2 + · · ·+ (∥ − (i, i)∥+ + ∥(i, i)∥−)2

)1/2
=
(
(∥(i, i)∥+ + ∥(i, i)∥−)2 + · · ·+ (∥(i, i)∥+ + ∥(i, i)∥−)2

)1/2
≥
(
∥(i, i)∥2J + · · ·+ ∥(i, i)∥2J

)1/2
=
(
(
√
2)2 + · · ·+ (

√
2)2
)1/2

= (2 + · · ·+ 2)1/2 =
√
2n

Note that a partition of the interval [
√
3, 4] was constructed, starting at

√
3, then alter-

nating between rational and irrational numbers until it ends at 4, for which

2

V4√
3(f, (C2, [·, ·]))

is not finite, thus f is not of bounded 2-variation in (C2, [·, ·]).

Theorem 9. Let (K = K+
˙[+]K−, [·, ·]) be a Krein space, f : [a, b] → K, is of bounded

q-variation in [a, b] on (K = K+
˙[+]K−), then f is of bounded q-variation in [a, b] on the

Hilbert spaces (K+, [·, ·]) and (K−,−[·, ·]).

Proof. Consider the partition P = {a, t1, t2, · · · , ti−1, ti, · · · , tn−1, b} ∈ P[a, b] y q ≥ 1.

The proof is a consequence of:
∥f−(ti)−f−(ti−1)∥q−, ∥f+(ti)−f+(ti−1)∥q+ ≤ (∥f+(ti)−f+(ti−1)∥++∥f−(ti)−f−(ti−1)∥−)q



O. Ferrer, J. Naranjo / Eur. J. Pure Appl. Math, 18 (2) (2025), 5565 11 of 18

Theorem 10. Let (K = K+
˙[+]K−, [·, ·]) be a Krein space and f : [a, b] → K = K+

˙[+]K−

a function, f ∈ B
q

V ([a, b],K), then
q+

Vb
a(f, (K+, [·, ·])) = 0 =

q−

Vb
a(f, (K−,−[·, ·])) if and only

if f is constant in [a, b] with respect to (K = K+
˙[+]K−).

Proof.

(→) Suppose that

q+

Vb
a(f, (K+, [·, ·])) = 0 =

q−

Vb
a(f, (K−,−[·, ·])), that is,

q+

Vb
a(f, (K+, [·, ·])) = sup

P∈P[a,b]


(

n∑
i=1

∥f+(ti)− f+(ti−1)∥q+

)1/q
 = 0

and

q−

Vb
a(f, (K−,−[·, ·])) = sup

P∈P[a,b]


(

n∑
i=1

∥f−(ti)− f−(ti−1)∥q−

)1/q
 = 0.

Then,(
n∑

i=1

∥f+(ti)− f+(ti−1)∥q+

)1/q

= 0 and

(
n∑

i=1

∥f−(ti)− f−(ti−1)∥q−

)1/q

= 0.

Let x ∈ [a, b]. Then, in particular, for the partition P = {a, x, b}, we have that(
∥f+(x)− f+(a)∥q+ + ∥f+(b)− f+(x)∥q+

)1/q
= 0 and(

∥f−(x)− f−(a)∥q− + ∥f−(b)− f−(x)∥q−
)1/q

= 0.

Whence,
∥f+(x)−f+(a)∥q++∥f+(b)−f+(x)∥q+ = 0, ∥f−(x)−f−(a)∥q−+∥f−(b)−f−(x)∥q− = 0.

Thus,

∥f+(x)− f+(a)∥q+ = 0, ∥f+(b)− f+(x)∥q+ = 0, ∥f−(x)− f−(a)∥q− = 0,
∥f−(b)− f−(x)∥q− = 0. Therefore,(
f+(x)−f+(a) = 0, f+(b)−f+(x) = 0

)
and

(
f−(x)−f−(a) = 0, f−(b)−f−(x) = 0

)
.

Next, (
f+(x) = f+(a), f+(b) = f+(x)) and (f−(x) = f−(a), f−(b) = f−(x)

)
Follows,

f+(x) = f+(a) = f+(b) and f−(x) = f−(a) = f−(b).

Therefore,

f(x) = f+(x) + f−(x) = f+(a) + f−(a) = f+(b) + f−(b).

Thus, f is constant in the interval [a, b].



O. Ferrer, J. Naranjo / Eur. J. Pure Appl. Math, 18 (2) (2025), 5565 12 of 18

(←) Suppose that f is constant in [a, b], then exists c ∈ K such that for any x ∈ [a, b],
f(x) = c. Since c ∈ K = K+

˙[+]K−, there are c+ ∈ K+ and c− ∈ K− such that
c = c+ + c−. Furthermore, as f(x) = f+(x) + f−(x), it follows that:

f+(x) + f−(x) = c+ + c−, whence f+(x)− c+ = c− − f−(x) ∈ K+ ∩ K− = {0}

Thus,

f+(x)− c+ = 0 and c− − f−(x) = 0, therefore, f+(x) = c+ and c− = f−(x),

Thus,

q+

Vb
a(f, (K+, [·, ·])) = sup

P∈P[a,b]


(

n∑
i=1

∥f+(ti)− f+(ti−1)∥q+

)1/q


= sup
P∈P[a,b]


(

n∑
i=1

∥c+ − c+∥q−

)1/q
 = sup{0} = 0

and

q−

Vb
a(f, (K−,−[·, ·])) = sup

P∈P[a,b]


(

n∑
i=1

∥f−(ti)− f−(ti−1)∥q−

)1/q


= sup
P∈P[a,b]


(

n∑
i=1

∥c− − c−∥q−

)1/q
 = sup{0} = 0

5.1. Algebra of bounded q-variation functions in Krein spaces

Theorem 11. Let (K = K+
˙[+]K−, [·, ·]) be a Krein space, if f, g : [a, b]→ K be of bounded

q-variation functions in [a, b] on (K = K+
˙[+]K−) and α a scalar, then αf and f + g are

also of bounded q-variation functions in [a, b] on (K = K+
˙[+]K−).

Proof. If f and g are functions of bounded q-variation in [a, b] on (K = K+
˙[+]K−, [·, ·]),

then exists L,M > 0 such that

q

Vb
a(f, (K, [·, ·])) ≤ L and

q

Vb
a(g, (K, [·, ·])) ≤M

(i) If α is a scalar,considering part two (i) and (iii) of the remark 4, for all partition
P ∈ P[a, b], it follows that:



O. Ferrer, J. Naranjo / Eur. J. Pure Appl. Math, 18 (2) (2025), 5565 13 of 18

q

Vb
a(αf, (K, [·, ·]))

= sup
P∈P[a,b]

{(
n∑

i=1

(
∥αf+(ti)− αf+(ti−1)∥+ + ∥αf−(ti)− αf−(ti−1)∥−

)q
)1/q}

= sup
P∈P[a,b]

{( n∑
i=1

(
∥α
(
f+(ti)− f+(ti−1))∥+ + ∥α(f−(ti)− f−(ti−1)

)
∥−
)q)1/q }

= sup
P∈P[a,b]

{( n∑
i=1

(
|α|∥f+(ti)− f+(ti−1)∥+ + |α|∥f−(ti)− f−(ti−1)∥−

)q)1/q }

= sup
P∈P[a,b]

{(
|α|q

n∑
i=1

(
∥f+(ti)− f+(ti−1)∥+ + ∥f−(ti)− f−(ti−1)∥−

)q)1/q }

= |α| sup
P∈P[a,b]

{( n∑
i=1

(
∥f+(ti)− f+(ti−1)∥+ + ∥f−(ti)− f−(ti−1)∥−

)q)1/q }
= |α|

q

Vb
a(f, (K, [·, ·])) ≤ |α|L.

Therefore,

q

Vb
a(αf, (K, [·, ·])) is finite, which implies that αf is of bounded q-variation

in [a, b] on (K = K+
˙[+]K−, [·, ·]).

(ii) Using (ii) and (iv) in part two of remark 4 it follows that for any partition P ∈ P[a, b],
it is satisfied that:

(
n∑

i=1

(
∥(f + g)+(ti)− (f + g)+(ti−1)∥+ + ∥(f + g)−(ti)− (f + g)−(ti−1)∥−

)q)1/q

=

(
n∑

i=1

(∥f+(ti)− f+(ti−1) + g+(ti)− g+(ti−1)∥+ + ∥f−(ti)− f−(ti−1) + g−(ti)− g−(ti−1)∥−)q)1/q

≤

(
n∑

i=1

((
∥f+(ti)− f+(ti−1)∥+ + ∥f−(ti)− f−(ti−1∥−

)
+

(
∥g+(ti)− g+(ti−1)∥+ + ∥g−(ti)− g−(ti−1)∥−

))q)1/q

Then by the Minkowski inequality [7], it follows that:(
n∑

i=1

(
∥(f + g)+(ti)− (f + g)+(ti−1)∥+ + ∥(f + g)−(ti)− (f + g)−(ti−1)∥−

)q)1/q



O. Ferrer, J. Naranjo / Eur. J. Pure Appl. Math, 18 (2) (2025), 5565 14 of 18

≤

(
n∑

i=1

(
∥f+(ti)− f+(ti−1)∥+ + ∥f−(ti)− f−(ti−1)∥−

)q)1/q

+

(
n∑

i=1

(
∥g+(ti)− g+(ti−1)∥+ + ∥g−(ti)− g−(ti−1)∥−

)q)1/q

Whence,

q

Vb
a((f + g), (K, [·, ·])) ≤

q

Vb
a(f, (K, [·, ·])) +

q

Vb
a(g, (K, [·, ·])) ≤ L+M.

Therefore, f+g is of bounded q-variation in [a, b] on the Krein space (K = K+
˙[+]K−, [·, ·]).

Theorem 12. Let (K = K+
˙[+]K−, [·, ·]) be a Krein space and f : [a, b] → K = K+

˙[+]K−

a function, if f ∈ B
q

V b
a (K, [·, ·]), then f ∈ B

p

V b
a (K, [·, ·]) para p > q.

Proof. Suppose that f ∈
q

V b
a (K, [·, ·]), then existsM > 0 such that

q

Vb
a(f, (K, [·, ·])) ≤M

Now, for p > q, let us consider two cases:

i) ∥f+(ti)− f+(ti−1)∥+ + ∥f−(ti)− f−(ti−1)∥− ≥ 1. Then,

(∥f+(ti) − f+(ti−1)∥+ + ∥f−(ti) − f−(ti−1)∥−)q ≤ (∥f+(ti) − f+(ti−1)∥+ + ∥f−(ti) −
f−(ti−1)∥−)p. Later,∑n

i=1(∥f+(ti)− f+(ti−1)∥+ + ∥f−(ti)− f−(ti−1)∥−)q ≤
∑n

i=1(∥f+(ti)− f+(ti−1)∥+ +
∥f−(ti)− f−(ti−1)∥−)p

Since p > q, then
1

p
<

1

q
and thus:

p

Vb
a(f, (K, [·, ·])) = sup

P∈P[a,b]


(

n∑
i=1

(
∥f+(ti)− f+(ti−1)∥+ + ∥f−(ti)− f−(ti−1)∥−

)p)1/p


≤ sup
P∈P[a,b]


(

n∑
i=1

(
∥f+(ti)− f+(ti−1)∥+ + ∥f−(ti)− f−(ti−1)∥−

)q)1/q


≤M

ii) 0 ≤ ∥f+(ti)− f+(ti−1)∥+ + ∥f−(ti)− f−(ti−1)∥− < 1. Then

(∥f+(ti)− f+(ti−1)∥+ + ∥f−(ti)− f−(ti−1)∥−)p ≤ (∥f+(ti)− f+(ti−1)∥+ + ∥f−(ti)−
f−(ti−1)∥−)q. Whence,



O. Ferrer, J. Naranjo / Eur. J. Pure Appl. Math, 18 (2) (2025), 5565 15 of 18∑n
i=1(∥f+(ti)− f+(ti−1)∥+ + ∥f−(ti)− f−(ti−1)∥−)p ≤

∑n
i=1(∥f+(ti)− f+(ti−1)∥+ +

∥f−(ti)− f−(ti−1)∥−)q

and therefore,

(
n∑

i=1

(
∥f+(ti)− f+(ti−1)∥+ + ∥f−(ti)− f−(ti−1)∥−

)p)1/p

is less than or equal to

(
n∑

i=1
(∥f+(ti)− f+(ti−1)∥+ + ∥f−(ti)− f−(ti−1)∥−)q

)1/q

,

it follows that:

p

Vb
a(f, (K, [·, ·])) ≤

q

Vb
a(f, (K, [·, ·]))

From which it follows for cases i) y ii) that

p

Vb
a(f, (K, [·, ·])) is finite, therefore

f ∈ B
p

V b
a (K, [·, ·]).

Next, the set of bounded q−variation functions on Krein space, que which we introduce
in 7 will be given a norm, which we will call q−norm and denote by ∥ · ∥q

Theorem 13. Let (K = K+
˙[+]K−, [·, ·]) be a Krein space, the function

|| · ||q : B
q

V b
a (K, [·, ·])→ R

defined by

∥f∥q = ∥f+(x)∥+ + ∥f−(x)∥− +

q+

Vb
a(f, (K+, [·, ·])) +

q−

Vb
a(f, (K−,−[·, ·])),

is a norm in B
q

V b
a (K, [·, ·]).

Proof. Let f, g ∈ B
q

V b
a (K, [·, ·]), λ ∈ R.

If x ∈ [a, b], then we have:

(i) ∥f+(x)∥+ ≥ 0, ∥f−(x)∥− ≥ 0. Also satisfied that

q+

Vb
a(f, (K+, [·, ·])) ≥ 0 and

q−

Vb
a(f, (K−,−[·, ·])) ≥ 0. Therefore, ∥f∥

B
q

V b
a (K,[·,·])

≥ 0.

(ii) If ∥f∥
B

q

V b
a (K,[·,·])

= 0, then

q+

V b
a (f, (K+, [·, ·])) +

q−

V b
a (f, (K−,−[·, ·])) + ∥f+(x)∥+ + ∥f−(x)∥− = 0.



O. Ferrer, J. Naranjo / Eur. J. Pure Appl. Math, 18 (2) (2025), 5565 16 of 18

Since
q+

V b
a (f, (K+, [·, ·])),

q−

V b
a (f, (K−,−[·, ·])), ∥f+(x)∥+, ∥f−(x)∥− ≥ 0, then

q+

V b
a (f, (K+, [·, ·])) =

q−

V b
a (f, (K−,−[·, ·])) = ∥f+(x)∥+ = ∥f−(x)∥− = 0. (2)

As the positive and negative variations are zero, then by theorem 10 f is constant
in K+ and in K−, so there exist c+ ∈ K+ and c− ∈ K− such that:

f+(x) = c+ and f−(x) = c−, ∀x ∈ [a, b]

Then, using the above and equation 2 it follows that:

∥f+(x)∥+ = ∥c+∥+ = 0 and ∥f−(x)∥− = ∥c−∥− = 0

it implies that c+ = 0 and c− = 0. Therefore, f(x) = 0 + 0 = 0 for all x ∈ [a, b],
then f is the null function.

(iii) Since f ∈ B
q

V b
a (K, [·, ·]) and λ ∈ C, using (i) and (iii) of the remark 4, then

∥λf∥
B

q

V b
a (K,[·,·])

= ∥λf+(x)∥+ + ∥λf−(x)∥− +
q+

V b
a (λf, (K+, [·, ·])) +

q−

V b
a (λf, (K−,−[·, ·]))

= |λ|∥f+(x)∥+ + |λ|∥f−(x)∥− + |λ|
q+

V b
a (f, (K+, [·, ·])) + |λ|

q−

V b
a (f, (K−,−[·, ·]))

= |λ|(∥f+(x)∥+ + ∥f−(x)∥− +
q+

V b
a (f, (K+, [·, ·])) +

q−

V b
a (f, (K−,−[·, ·])))

= |λ|∥f∥
B

q

V b
a (K,[·,·])

.

(iv) (Triangular inequality )

Let f, g ∈ B
q

V b
a (K, [·, ·]), using (ii) and (iv) of the remark 4, it follows that:

∥f + g∥
B

q

V b
a (K,[·,·])

= ∥(f + g)+(x)∥+ + ∥(f + g)−(x)∥− +
q+

V b
a (f + g, (K+, [·, ·])) +

q−

V b
a (f + g, (K−,−[·, ·]))

≤ ∥f+(x)∥+ + ∥g+(x)∥+ + ∥f−(x)∥− + ∥g−(x)∥− +
q+

V b
a (f, (K+, [·, ·]))+

q+

V b
a (g, (K+, [·, ·])) +

q−

V b
a (f, (K−,−[·, ·])) +

q−

V b
a (g, (K−,−[·, ·]))

= ∥f+(x)∥+ + ∥f−(x)∥− +
q+

V b
a (f, (K+, [·, ·])) +

q−

V b
a (f, (K−,−[·, ·])) + ∥g+(x)∥++

∥g−(x)∥− +
q+

V b
a (g, (K+, [·, ·])) +

q−

V b
a (g, (K−,−[·, ·]))

= ∥f∥
B

q

V
b

a(K,[·,·])
+ ∥g∥

B
q

V
b

a(K,[·,·])



O. Ferrer, J. Naranjo / Eur. J. Pure Appl. Math, 18 (2) (2025), 5565 17 of 18

Thus, ∥f∥
B

q

V b
a (K,[·,·])

= ∥f+(x)∥+ + ∥f−(x)∥− +
q+

V b
a (f, (K+, [·, ·])) +

q−

V b
a (f, (K−,−[·, ·])) is a

norm in B
q

V b
a (K, [·, ·]).

6. Conclusion and future work

In this study, the concept of bounded q- variation function in Krein spaces was defined
(definition 7 ) and exemplified (example 3) , extending the existing notion of these func-
tions in Hilbert spaces. Classical results were extended (theorem 6, theorem 7, theorem 8,
theorem 9, theorem 10, theorem 11, theorem 12, theorem 13 ), showing the potential that
this research has for further extensions and applications. Future research could focus on
extending work presented in [18] to spaces with indefinite metrics. In addition, investigate
the interaction between bounded q-variational functions on spaces with indefinite metrics
and fixed point theory. Knowing the importance of spaces of indefinite metric [19], in
quantum mechanics [15], introducing a q-norm (theorem 13) for bounded variation func-
tions in spaces with an indefinite metric can provide applications such as those given in
which gives insight into the future impact of this work.

Acknowledgements

The authors are gratefully acknowledged for contributing equally to the manuscript.
The authors thank the Universities of Sucre and Pontificia Bolivariana for their support.
Strong thanks are also due to the editor and referees of the journal for their helpful
suggestions that allowed us to improve this manuscript.

Conflict of interest

The authors declare that they have no conflict of interest in this work.

References

[1] V. V. Chistyakov. On mappings of bounded variation. J Dyn Control Syst, 2:261–289,
1997.

[2] V. V. Chistyakov. On the theory of multivalued mappings of bounded variation of
one real variable. Sb Math, 189:153–176, 1998.

[3] V. V. Chistyakov. Metric-valued mappings of bounded variation. J Math Sci (N.Y.),
111:3387–3429, 2002.

[4] O. Ferrer, J. Naranjo, and C. Guzmán. Strongly bounded variation in krein spaces.
Journal of Mathematics and Computer Science, 36(2):237–250, 2024.

[5] J. Cure, K. Ferrer, and O. Ferrer. Functions of bounded (2, k)-variation in 2-normed
spaces. AIMS Mathematics, 9(9):24166–24183, 2024.



O. Ferrer, J. Naranjo / Eur. J. Pure Appl. Math, 18 (2) (2025), 5565 18 of 18

[6] J. Brokman, M. Burger, and G. Gilboa. Spectral total-variation processing of shapes-
theory and applications. ACM Transactions on Graphics, 43(2), 2024.

[7] M. Di Francesco. Functional Analysis in Applied Mathematics and Engineering. 2019.
[8] D. Bugajewska, D. Bugajewski, and H. Hudzik. bvϕ-solutions of nonlinear integral

equations. Journal of Mathematical Analysis and Applications, 287(1):265–278, 2003.
[9] D. Bugajewska, D. Bugajewski, and G. Lewicki. On nonlinear integral equations in

the space of functions of bounded generalized ϕ-variation. The Journal of Integral
Equations and Applications, 21(1):1–20, 2009.

[10] X. Xie, Y. Liu, P. Li, and J. Huang. The bounded variation capacity and sobolev-type
inequalities on dirichlet spaces. Advances in Nonlinear Analysis, 13(1), 2024.

[11] T. Azizov and I. Iokhvidov. Linear operators in spaces with an indefinite metric. John
Wiley and Sons Ltd, Chichester, 1989.

[12] J. Bognar. Indefinite inner product spaces. Springer, Berlin, 1974.
[13] N. Weiner. The quadratic variation of function and its fourier coefficients. Mas-

sachusett J. Math. Phys., 3:72–94, 1924.
[14] Y. Abramovich, E. Avgerinos, and A.C. Yannelis. Functional Analysis and Economic

Theory. Springer-Verlag, Berlin, Heidelberg, 1998.
[15] P. A. M. Dirac. The physical interpretation of the quantum dynamics. Proceedings

of the Royal Society of London. Series A, Containing Papers of a Mathematical and
Physical Character, 113(765):621–641, 1927.

[16] J. Krvavych. Stock price modelling by long-memory processes: overview of the frac-
tional brownian approach. In Actuarial Studies Research Symposium, UNSW, Nov.,
2002.

[17] S. Osher, A. Solé, and L. Vese. Image decomposition, image restoration, and texture
modeling using total variation minimization and the h−1 norm. ICIP, 2003.

[18] P. Debnath, N. Konwar, and S. Radenović. Metric fixed point theory. Springer,
Singapore, 2021.

[19] M.Langer and A. Luger. On norms in indefinite inner product spaces. in: Recent
advances in operator theory in Hilbert and Krein spaces. Oper. Theory Adv. Appl,
2010.