EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 4, Article Number 6613
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Tensor Product of Spaces with Generalized 2-Inner
Product

Moreno Luis1, Ferrer Osmin1,∗, Sierra Arley1,2

1 University of Sucre, Faculty of Education and Sciences, Department of Mathematics,
Sincelejo, Colombia
2 Caribbean University Corporation, Department of Basic Sciences, Sincelejo, Colombia

Abstract. In this work, we introduce the notion of tensor product of spaces with a generalized
2-inner product (see Definition 5), and we establish several interesting properties (see Proposition
5), thereby generalizing the classical properties of the tensor product of inner product spaces.
Moreover, we equip this tensor product with a mapping that defines a generalized 2-inner product
(see Theorem 3) and, consequently, endow it with a generalized 2-norm (see Theorem 1). In this
context, we also define the tensor product of linear operators (see Definition 9) and prove a series of
results for example, that the tensor product of two 2-bounded linear operators is again 2-bounded
under the tensor product (see Proposition 10).

2020 Mathematics Subject Classifications: 46M05, 47A80, 46B28, 46C50

Key Words and Phrases: Tensor product, generalized 2-inner product, 2-norm

1. Introduction

The ideas that gave rise to the concept of the tensor product of vector spaces were
developed by various researchers throughout the nineteenth century; however, it was not
until the work carried out by the mathematician Hassler Whitney (1938) that the notion of
the tensor product of abelian groups and more generally of modules was firmly established
[1]. Consequently, the tensor product of vector spaces was cast in the language of universal
properties in works such as those of Bourbaki (1943) in the article [2] and in the book
by Artin, Nesbitt, and Thrall (1944) [3]. These developments, in turn, spurred rapid
advancement and further elaboration of the concept in other mathematical contexts.

The notion of the tensor product of modules has been extensively developed in various
mathematical contexts for example, within Homological Algebra and Differential Geometry
(see [4–8]). It is a concept of great significance, since, broadly speaking, it provides a

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6613

Email addresses: morenoarroyoluismario@gmail.com (M. Luis),
osmin.ferrer@unisucre.edu.co (F. Osmin), arleysierra23@gmail.com (S. Arley)

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



M. Luis, F. Osmin, S. Arley / Eur. J. Pure Appl. Math, 18 (4) (2025), 6613 2 of 17

method for constructing new spaces with the desired structure and, in the case of vector
spaces, offers a way to pass from bilinear operators to linear operators. Moreover, it arises
naturally in areas of physics such as quantum physics and quantum computing (see [9, 10]).

In recent years, the concept of the tensor product of complex vector spaces equipped
with a positive definite sesquilinear form [11] has been extended to the tensor product
of spaces endowed with an arbitrary sesquilinear form [12], defining simple tensors using
such sesquilinear forms in an appropriate manner. Moreover, since the concept of the
classical inner product can be extended to what is known as the 2-inner product and the
generalized 2-inner product in the senses of Gähler [13] and Lewandoska [14], respectively,
it is natural to consider defining the tensor product of spaces endowed with a 2-inner
product of this kind, which in this work will be referred to as the 2-tensor product of
spaces with a generalized 2-inner product. The theory of spaces with a 2-inner product
has been extensively studied in articles by Diminnie, Gähler and White [15–18].

Thus, the present work is organized as follows: in Section 2, we include the necessary
preliminaries to introduce the notion of the algebraic 2-tensor product of vector spaces
endowed with a generalized 2-inner product, such as the concept of a generalized 2-inner
product and some of its properties. In Section 3, we introduce the notion of the 2-tensor
product of elements in spaces with a generalized 2-inner product, based on the ideas of the
simple tensor product of elements in spaces with a classical inner product; consequently, we
establish the notion of the 2-tensor product of spaces with a generalized 2-inner product,
define a generalized 2-inner product on it, and prove some interesting properties. Section
4 contains the notion of the 2-tensor product of linear operators and some of their satisfied
properties. Finally, Section 5 presents the conclusions of this research.

2. Preliminaries

In this section, we study the concepts of the generalized 2-norm, the generalized 2-inner
product, and some of the most important properties that are satisfied in spaces endowed
with these structures.

Definition 1. [14] A generalized 2-norm on X is a map ∥·, ·∥ : X × X → R such that
for every w, y, z ∈ X is true

i) ∥w, z∥ = ∥z, w∥;

ii) ∥w,αz∥ = |α|∥w, z∥ for all α ∈ C;

iii) ∥w, y + z∥ ≤ ∥w, y∥+ ∥w, z∥.

The pair (X , ∥·, ·∥) is called a space with generalized 2-norm.

In the following example we show how to define a generalized 2-norm from an indefinite
sesquilinear form.



M. Luis, F. Osmin, S. Arley / Eur. J. Pure Appl. Math, 18 (4) (2025), 6613 3 of 17

Example 1. [19] We define on the vector space Cn the function ∥·, ·∥ : Cn ×Cn → R by
means of the rule ∥x, y∥ :=

∣∣∑n
i=1(−1)ixiyi

∣∣ for all x = (x1, · · · , xn), y = (y1, · · · , yn) ∈
Cn. What defines a generalized 2-norm. In general, it can be shown that, given a classical
inner product space (X , ⟨·, ·⟩), the mapping

∥·, ·∥ : X × X → R
(w, z) 7→ ∥w, z∥ = |⟨w, z⟩|,

defines a generalized 2-norm on X .

Below we present the concept of generalized 2-inner product, a notion of great importance
for this work.

Definition 2. [14] A generalized 2-inner product is a map ⟨·, ·|·⟩ : X ×X ×X −→ C such
that for all w, y, z, w1, w2 ∈ X and for all α ∈ C

i) ⟨y, w|z⟩ = ⟨w, y|z⟩;

ii) ⟨w1 + w2, y|z⟩ = ⟨w1, y|z⟩+ ⟨w2, y|z⟩;

iii) ⟨αw, y|z⟩ = α⟨w, y|z⟩;

iv) ⟨w,w|z⟩ = ⟨z, z|w⟩;

v) ⟨w,w|z⟩ ≥ 0.

A complex vector space X together a generalized 2-inner product ⟨·, ·|·⟩, denoted by
(X , ⟨·, ·|·⟩), it is said to be a generalized 2-inner product space.

Proposition 1. [20]It is easy to check that the following function is a generalized 2-inner
product on X

⟨w, y|z⟩ =
∣∣∣∣⟨w, y⟩ ⟨w, z⟩
⟨z, y⟩ ⟨z, z⟩

∣∣∣∣ = ⟨w, y⟩ ⟨z, z⟩ − ⟨w, z⟩ ⟨z, y⟩ for all w, y, z ∈ X .

This 2-inner product is called generalized standard 2-inner product and it is denoted by
⟨·, ·|·⟩stand.

A consequence of [21, Prop 3.1] when (X , ⟨·, ·⟩) is a classical inner product space is:

Proposition 2. [21] Let (X , ⟨·, ·⟩) be a classical inner product space. Then the mapping

⟨·, ·|·⟩ : X × X × X −→ C

defined by
⟨x, y|z⟩ = ⟨x, y⟩ ∥z∥2 for all x, y, z ∈ X ,

is a generalized 2-inner product on X .



M. Luis, F. Osmin, S. Arley / Eur. J. Pure Appl. Math, 18 (4) (2025), 6613 4 of 17

Proposition 3. [19] Let (X , ⟨·, ·|·⟩) be a generalized 2-inner product space. Then, for all
w, y, z ∈ X it holds that:

|⟨w, y|z⟩|2 ≤ ⟨w,w|z⟩⟨y, y|z⟩.

Remark 1. The function

∥w, z∥ :=
√

⟨w,w|z⟩, w, z ∈ X ,

sometimes it is called induced generalized 2-norm by the generalized 2-inner product.

Definition 3 (2-bounded operator). [14] Let X be a vector space endowed with two
generalized 2-norms ∥·, ·∥1 and ∥·, ·∥2. An operator T : (X, ∥·, ·∥1) → (X, ∥·, ·∥2) is said to
be 2-bounded if there α ≥ 0 such that

∥T (x), y∥2 + ∥x, T (y)∥2 ≤ α∥x, y∥1 for all x, y ∈ X .

The symbol
2
B(X ) will denote the set of 2-bounded linear operator on X , that is,

2
B(X ) := {T : X → X : T is linear and 2-bounded}.

Remark 2. It is clear that the set
2
B(X ) can be endowed with a vector space structure

over C, by means of the pointwise operations of operators.

By virtue of the properties of generalized 2-normed spaces, in the following theorem
we establish an equivalent result for the Definition 3.

Theorem 1. Note that a linear operator T : (X , ∥·, ·∥1) → (X , ∥·, ·∥2) is 2-bounded according
to Definition 3 if and only if there exists a constant β > 0 such that

∥Tz,w∥2 ≤ β ∥z, w∥1 for all z, w ∈ X .

Proof. Suppose first that there exists α > 0 such that

∥Tx, y∥2 + ∥x, Ty∥2 ≤ α ∥x, y∥1 for all x, y ∈ X .

Since each 2-norm is nonnegative, it follows that

∥Tx, y∥2 ≤ ∥Tx, y∥2 + ∥x, Ty∥2 ≤ α ∥x, y∥1,

hence T satisfies the above inequality with β = α. Conversely, assume there exists β > 0
such that

∥Tz,w∥2 ≤ β ∥z, w∥1 for all z, w ∈ X .

Then for any x, y ∈ X we have

∥Tx, y∥2 + ∥x, Ty∥2 = ∥Tx, y∥2 + ∥Ty, x∥2 ≤ β ∥x, y∥1 + β ∥y, x∥1 = 2β ∥x, y∥1.

If we set α = 2β, the desired inequality follows.



M. Luis, F. Osmin, S. Arley / Eur. J. Pure Appl. Math, 18 (4) (2025), 6613 5 of 17

Definition 4. If T ∈
2
B(X ), we define ∥T∥, by

∥T∥ = inf{a ≥ 0 : ∥T (w), z∥ ≤ a∥w, z∥ for all w, z ∈ X}.

Note that the identity operator in X , I : X → X , fulfills ∥I∥ = 1.

Theorem 2. In the context of the Definition 4, for all T ∈
2
B(X ) it is true that

∥T∥ = sup{∥Tw, z∥ : w, z ∈ X and ∥w, z∥ = 1}
∥T∥ = sup{∥Tw, z∥ : w, z ∈ X and ∥w, z∥ ≤ 1}

∥T∥ = sup

{
∥Tw, z∥
∥w, z∥

: w, z ∈ X and ∥w, z∥ ≠ 0

}
Moreover, thanks to the Definition 4 we prove that given a generalized 2-normed space

X , it is possible to endow the vector space
2
B(X ) with the structure of a semi-normed

space.

Proposition 4. [14] For all T ∈
2
B(X ) it is true that

∥T (w), y∥ ≤ ∥T∥∥w, y∥

for all w, y, z ∈ X .

3. Main Results

3.1. 2-Tensor Product

In this section, we introduce the notion of the 2-tensor product of elements in vector
spaces equipped with a generalized 2-inner product.

Definition 5. Let (X1, ⟨·, ·|·⟩1) and (X2, ⟨·, ·|·⟩2) be spaces with a generalized 2-inner
product. Given x1 ∈ X1 and x2 ∈ X2, the 2-tensor product of x1 and x2, denoted by

x1
2
⊙ x2, is the function

x1
2
⊙ x2 : (X1 ×X2)× (X1 ×X2) −→ C

defined by

(x1
2
⊙ x2)

(
(t1, t2), (r1, r2)

)
= ⟨x1, t1 | r1⟩1 ⟨x2, t2 | r2⟩2.

Proposition 5. Let (X1, ⟨·, ·|·⟩1) and (X2, ⟨·, ·|·⟩2) be spaces with a generalized 2-inner
product. For all x, x1, y1 ∈ X1, x2, y2, y ∈ X2, and all α, β ∈ C, the following properties
hold:



M. Luis, F. Osmin, S. Arley / Eur. J. Pure Appl. Math, 18 (4) (2025), 6613 6 of 17

i) x
2
⊙ 02 = 0

ii) 01

2
⊙ y = 0

iii) (αx1)
2
⊙ x2 = α(x1

2
⊙ x2) = x1

2
⊙ (αx2)

iv) αβ(x1
2
⊙ x2) = (αx1

2
⊙ βx2)

v) (x1 + y1)
2
⊙ x2 = (x1

2
⊙ x2) + (y1

2
⊙ x2)

vi) x1
2
⊙ (x2 + y2) = (x1

2
⊙ x2) + (x1

2
⊙ y2)

Proof. Let t1, r1 ∈ X1 be and t2, r2 ∈ X2 be.

i) (x1
2
⊙ 02)((t1, t2), (r1, r2)) = ⟨x1, t1|r1⟩1⟨02, t2|r2⟩2 = ⟨x1, t1|r1⟩1 · 0 = 0

ii) (01
2
⊙ x2)((t1, t2), (r1, r2)) = ⟨01, t1|r1⟩1⟨x2, t2|r2⟩2 = 0 · ⟨x2, t2|r2⟩2 = 0

iii) ((αx1)
2
⊙ x2)((t1, t2), (r1, r2)) = ⟨αx1, t1|r1⟩1⟨x2, t2|r2⟩2 = α⟨x1, t1|r1⟩1⟨x2, t2|r2⟩2

= ⟨x1, t1|r1⟩1⟨(αx2), t2|r2⟩2 = (x1
2
⊙ (αx2))((t1, t2), (r1, r2))

= α(x1
2
⊙ x2)((t1, t2), (r1, r2))

iv) (αx1
2
⊙ βx2)((t1, t2), (r1, r2)) = ⟨(αx1), t1|r1⟩1⟨(βx2), t2|r2⟩2 = α⟨x1, t1|r1⟩1β⟨x2, t2|r2⟩2

= αβ⟨x1, t1|r1⟩1⟨x2, t2|r2⟩2 = αβ(x1
2
⊙ x2)((t1, t2), (r1, r2))

v) ((x1 + y1)
2
⊙ x2)((t1, t2), (r1, r2)) = ⟨(x1 + y1), t1|r1⟩1⟨x2, t2|r2⟩2

= (⟨x1, t1|r1⟩1 + ⟨y1, t1|r1⟩1)⟨x2, t2|r2⟩2
= ⟨x1, t1|r1⟩1⟨x2, t2|r2⟩2 + ⟨y1, t1|r1⟩1⟨x2, t2|r2⟩2

= (x1
2
⊙ x2)((t1, t2), (r1, r2)) + (y1

2
⊙ x2)((t1, t2), (r1, r2))

vi) (x1
2
⊙ (x2 + y2))((t1, t2), (r1, r2)) = ⟨x1, t1|r1⟩1⟨(x2 + y2), t2|r2⟩2

= ⟨x1, t1|r1⟩1(⟨x2, t2|r2⟩2 + ⟨y2, t2|r2⟩2)
= ⟨x1, t1|r1⟩1⟨x2, t2|r2⟩2 + ⟨x1, t1|r1⟩1⟨y2, t2|r2⟩2

= (x1
2
⊙ x2)((t1, t2), (r1, r2)) + (x1

2
⊙ y2)((t1, t2), (r1, r2))



M. Luis, F. Osmin, S. Arley / Eur. J. Pure Appl. Math, 18 (4) (2025), 6613 7 of 17

3.2. Algebraic Tensor Product

Next, we introduce the notion of the algebraic tensor product between two vector
spaces endowed with a generalized 2-inner product.

Definition 6. Let (X1, ⟨·, ·|·⟩1) and (X2, ⟨·, ·|·⟩2) be spaces with a generalized 2-inner

product. We define their algebraic tensor product, denoted by X1

2
⊙X2, as

X1

2
⊙X2 :=

{ n∑
i=1

βi (xi
2
⊙ yi)

∣∣ n ∈ N∗, βi ∈ C, xi ∈ X1, yi ∈ X2

}
.

Note that X1

2
⊙X2 is a complex vector space.

Note that by the properties of the mapping
2
⊙ every element of the vector spaceX1

2
⊙X2

can be written simply as
∑n

i=1 xi
2
⊙ yi with xi ∈ X1, yi ∈ X2 and n ∈ N.

In the following theorem, we equip the vector space from Definition 6 with a generalized
2-inner product, which we call the generalized 2-inner tensor product.

Theorem 3. Let (X1, ⟨·, ·|·⟩1) and (X2, ⟨·, ·|·⟩2) be spaces with a generalized 2-inner product.
Then the mapping

⟨ξ, η | λ⟩ 2
⊙

: (X1

2
⊙X2)× (X1

2
⊙X2)× (X1

2
⊙X2) −→ C

defined by

〈
ξ, η | λ

〉
2
⊙
=

〈 n∑
i=1

xi
2
⊙ yi,

m∑
j=1

zj
2
⊙wj

∣∣∣ p∑
t=1

rt
2
⊙ st

〉
:=

n∑
i=1

m∑
j=1

p∑
t=1

δi,j ⟨xi, zj | rt⟩1 ⟨yi, wj | st⟩2,

where

ξ =

n∑
i=1

xi
2
⊙ yi, η =

m∑
j=1

zj
2
⊙ wj , λ =

p∑
t=1

rt
2
⊙ st,

is a generalized 2-inner product.

Proof. Let

ξ =
n∑

i=1

xi
2
⊙ yi, η =

m∑
j=1

zj
2
⊙ wj , λ =

p∑
t=1

rt
2
⊙ st, σ =

g∑
k=1

lk
2
⊙ vk

be elements of X1

2
⊙ X2 and let α ∈ C. To verify linearity in the first slot, first observe

that

λ =

p∑
t=1

rt
2
⊙ st =

n+p∑
i=n+1

xi
2
⊙ yi,



M. Luis, F. Osmin, S. Arley / Eur. J. Pure Appl. Math, 18 (4) (2025), 6613 8 of 17

where we set xn+t

2
⊙ yn+t := rt

2
⊙ st for each 1 ≤ t ≤ p. Hence

ξ + λ =

n+p∑
i=1

xi
2
⊙ yi.

Therefore:

⟨ξ + λ, η|σ⟩ 2
⊙

=

〈
n+p∑
i=1

xi
2
⊙ yi,

m∑
j=1

zj
2
⊙ wj |

g∑
k=1

lk
2
⊙ vk

〉
2
⊙

=

n+p∑
i=1

m∑
j=1

g∑
k=1

δi,j⟨xi, zj |lk⟩1⟨yi, wj |vk⟩2

=

n∑
i=1

m∑
j=1

g∑
k=1

δi,j⟨xi, zj |lk⟩1⟨yi, wj |vk⟩2 +
n+p∑

i=n+1

m∑
j=1

g∑
k=1

δi,j⟨xi, zj |lk⟩1⟨yi, wj |vk⟩2

=

n∑
i=1

m∑
j=1

g∑
k=1

δi,j⟨xi, zj |lk⟩1⟨yi, wj |vk⟩2 +
p∑

t=1

m∑
j=1

g∑
k=1

δn+t,j⟨xn+t, zj |lk⟩1⟨yn+t, wj |vk⟩2

=

n∑
i=1

m∑
j=1

g∑
k=1

δi,j⟨xi, zj |lk⟩1⟨yi, wj |vk⟩2 +
p∑

t=1

m∑
j=1

g∑
k=1

δi,j⟨ri, zj |lk⟩1⟨si, wj |vk⟩2

= ⟨ξ, η|σ⟩ 2
⊙
+ ⟨λ, η|σ⟩ 2

⊙

In addition,

⟨αξ, η|λ⟩ 2
⊙
=

〈
α

n∑
i=1

xi
2
⊙ yi,

r∑
j=1

zj
2
⊙ wj |

p∑
t=1

rt
2
⊙ st

〉
2
⊙

=

〈
n∑

i=1

α(xi
2
⊙ yi),

r∑
j=1

zj
2
⊙ wj |

p∑
t=1

rt
2
⊙ st

〉
2
⊙

=

〈
n∑

i=1

(αxi)
2
⊙ yi,

r∑
j=1

zj
2
⊙ wj |

p∑
t=1

rt
2
⊙ st

〉
2
⊙

=

n∑
i=1

m∑
j=1

p∑
t=1

δi,j⟨(αxi), zj |rt⟩1⟨yi, wj |st⟩2

=

n∑
i=1

m∑
j=1

p∑
t=1

αδi,j⟨xi, zj |rt⟩1⟨yi, wj |st⟩2 = α

n∑
i=1

m∑
j=1

p∑
t=1

δi,j⟨xi, zj |rt⟩1⟨yi, wj |st⟩2

= α⟨ξ, η|λ⟩ 2
⊙
.

Let us now show that this mapping is Hermitian.

⟨ξ, η|λ⟩ 2
⊙
=

n∑
i=1

r∑
j=1

p∑
t=1

δi,j ⟨xi, zj |rt⟩1 ⟨yi, wj |st⟩2 =
n∑

i=1

r∑
j=1

p∑
t=1

δi,j ⟨zj , xi|rt⟩1 ⟨wj , yi|st⟩2

=
r∑

j=1

n∑
i=1

p∑
t=1

δi,j ⟨zj , xi|rt⟩1 ⟨wj , yi|st⟩2 = ⟨η, ξ|λ⟩ 2
⊙
.



M. Luis, F. Osmin, S. Arley / Eur. J. Pure Appl. Math, 18 (4) (2025), 6613 9 of 17

Therefore the mapping ⟨·, ·|·⟩ 2
⊙
is Hermitian. Moreover, we have

⟨ξ, ξ|λ⟩ 2
⊙
=

〈
n∑

i=1

xi
2
⊙ yi,

n∑
l=1

xl
2
⊙ yl

∣∣∣ p∑
t=1

rt
2
⊙ st

〉
2
⊙

=
n∑

i=1

n∑
l=1

p∑
t=1

δi,l ⟨xi, xl|rt⟩1 ⟨yi, yl|st⟩2 =
n∑

i=1

p∑
t=1

⟨xi, xi|rt⟩1 ⟨yi, yi|st⟩2

=
n∑

i=1

p∑
t=1

⟨rt, rt|xi⟩1 ⟨st, st|yi⟩2 =
p∑

j=1

p∑
t=1

n∑
i=1

δj,t ⟨rj , rt|xi⟩1 ⟨sj , st|yi⟩2

=

〈
p∑

j=1

rj
2
⊙ sj ,

p∑
t=1

rt
2
⊙ st

∣∣∣ n∑
i=1

xi
2
⊙ yi

〉
2
⊙

= ⟨λ, λ|ξ⟩ 2
⊙
.

Finally, it is clear that ⟨ξ, ξ|λ⟩ 2
⊙

≥ 0 for all ξ, λ ∈ X1

2
⊙ X2. Hence ⟨·, ·|·⟩ 2

⊙
defines a

generalized 2-inner product on X1

2
⊙ X2.

Example 2. Consider C2 equipped with the application ⟨·, ·|·⟩C2 : C2 × C2 × C2 −→ C
given by

⟨(x1, x2), (y1, y2)|(z1, z2)⟩C2 := x1y1|z1|2 + x2y2|z2|2, (x1, x2), (y1, y2), (z1, z2) ∈ C2.

In addition, let us consider C with the application ⟨·, ·|·⟩C : C× C× C −→ C given by

⟨x, y|z⟩C := xy|z|2, x, y, z ∈ C.

Thus, given (a, b) ∈ C2 y c ∈ C, the function (a, b)
2
⊙ c : (C2 × C)× (C2 × C) → C

(
(a, b)

2
⊙ c

)
(((x1, x2), d), ((y1, y2), e)) := ⟨(a, b), (x1, x2)|(y1, y2)⟩C2 · ⟨c, d|e⟩C

=
(
ax1|y1|2 + bx2|y2|2

) (
cd|e|2

)
= acx1d|y1e|2 + bcx2d|y2e|2

= ⟨(ac, bc), (x1d, x2d)|(y1e, y2e)⟩C2 ,

for all ((x1, x2), d), ((y1, y2), e) ∈ C2 × C. Now, the tensor product of the spaces with
generalized 2-inner product (C2, ⟨·, ·|·⟩C2) and (C, ⟨·, ·|·⟩C), in accordance with the Definition

6, is given by C2
2
⊙ C =

{∑n
i=1(xi, yi)

2
⊙ ci : n ∈ N, (xi, yi) ∈ C2, ci ∈ C, 1 ≤ i ≤ n

}
, and

the generalized 2-inner product of Theorem 3 ⟨·, ·|·⟩ : (C2
2
⊙C)× (C2

2
⊙C)× (C2

2
⊙C) → C,

has the form



M. Luis, F. Osmin, S. Arley / Eur. J. Pure Appl. Math, 18 (4) (2025), 6613 10 of 17

⟨ξ, η|λ⟩ 2
⊙
:=

n∑
i=1

m∑
j=1

p∑
k=1

δi,j⟨(xi, yi), (zj , wj)|(rk, sk)⟩C2⟨ci, dj |ek⟩C

=

n∑
i=1

m∑
j=1

p∑
k=1

δi,j⟨(xici, yici), (zidi, widi)|(rkek, skek)⟩C2

=

n∑
i=1

p∑
k=1

⟨(xici, yici), (zidi, widi)|(rkek, skek)⟩C2

=

n∑
i=1

p∑
k=1

xicizidi|rkek|2 + yiciwidi|skek|2

for all ξ =
n∑

i=1

(xi, yi)
2
⊙ ci, η =

m∑
j=1

(zj , wj)
2
⊙ dj , λ =

p∑
k=1

(rk, sk)
2
⊙ ek ∈ C2

2
⊙ C.

Next, we equip the algebraic tensor product space from Definition 6 with a generalized
2-norm, which is induced by the mapping defined in Theorem 3.

Theorem 4. Let (X1, ⟨·, ·|·⟩1) and (X2, ⟨·, ·|·⟩2) be spaces with a generalized 2-inner product.

We define a generalized 2-norm on X1

2
⊙ X2, called the induced 2-tensor norm by the

generalized 2-inner product of Theorem 3, as the mapping

∥·, ·∥ 2
⊙
: (X1

2
⊙X2)× (X1

2
⊙X2) −→ R

given by

∥ξ, λ∥ 2
⊙
:=
√

⟨ξ, ξ|λ⟩ 2
⊙
, ξ, λ ∈ X1

2
⊙X2.

Proof. Since ⟨·, ·|·⟩ 2
⊙

is a generalized 2-inner product, it follows that ∥·, ·∥ 2
⊙

defines a

generalized 2-norm. The proof is straightforward and analogous to the classical case.

Following the work of Lewandoska [22], we introduce the notion of a 2-bounded linear
operator on the 2-tensor product of spaces with a generalized 2-inner product.

Definition 7 (2-bounded operator). Let
(
X1

2
⊙ X2, ∥·, ·∥ 2

⊙

)
be the generalized 2-normed

space from Theorem 1, and let T be a linear operator T a linear operator on X1

2
⊙X2. Then

we said that T is a 2-bounded linear operator if there exists a positive number α > 0 such
that

∥Tw, z∥ 2
⊙
≤ α ∥w, z∥ 2

⊙
for all w, z ∈ X1

2
⊙X2.

The symbol B(X1

2
⊙ X2) will denote the set of 2-bounded linear operator on X1

2
⊙ X2, that

is,

B(X1

2
⊙X2) := {T : X1

2
⊙X2 → X1

2
⊙X2 : T is linear and 2-bounded}.



M. Luis, F. Osmin, S. Arley / Eur. J. Pure Appl. Math, 18 (4) (2025), 6613 11 of 17

Remark 3. It is clear that the set B(X1

2
⊙X2) can be endowed with a vector space structure

over C, by means of the pointwise operations of operators.

Definition 8. [14] If T ∈ B(X1

2
⊙X2), we define ∥T∥ 2

⊙
, by

∥T∥ 2
⊙
= inf{a ≥ 0 : ∥T (w), z∥ 2

⊙
≤ a∥w, z∥ 2

⊙
for all w, z ∈ X}.

Theorem 5. [14] In the context of the Definition 8, for all T ∈
2
B(X ) it is true that

∥T∥ 2
⊙
= sup{∥Tw, z∥ 2

⊙
: w, z ∈ X and ∥w, z∥ 2

⊙
= 1}

∥T∥ 2
⊙
= sup{∥Tw, z∥ 2

⊙
: w, z ∈ X and ∥w, z∥ 2

⊙
≤ 1}

∥T∥ 2
⊙
= sup

∥Tw, z∥ 2
⊙

∥w, z∥ 2
⊙

: w, z ∈ X and ∥w, z∥ 2
⊙
̸= 0


Moreover, thanks to the Definition 8 we prove that given a generalized 2-normed space

X1

2
⊙ X2, it is possible to endow the vector space B(X1

2
⊙ X2) with the structure of a

semi-normed space.

Proposition 6. The mapping ∥ · ∥ 2
⊙
: B(X1

2
⊙X2) → R given by

∥T∥ 2
⊙
= sup

∥T (w), z∥ 2
⊙

∥w, z∥ 2
⊙

: w, z ∈ X1

2
⊙X2 and ∥w, z∥ 2

⊙
̸= 0


defines a semi-norm in B(X1

2
⊙X2).

The proofs are obtained from [14] using Theorem 1.

Proposition 7. [14] For all T ∈ B(X1

2
⊙X2) it is true that ∥T (w), y∥ 2

⊙
≤ ∥T∥∥w, y∥ 2

⊙
for

all w, y, z ∈ X1

2
⊙X2.

4. Tensor Product of Linear Operators

In the following definition, we establish the notion of the 2-tensor product of linear
operators on spaces with a generalized 2-inner product.

Definition 9. Let (X1, ⟨·, ·|·⟩1) and (X2, ⟨·, ·|·⟩2) be spaces with a generalized 2-inner
product, and let T : X1 → X1, S : X2 → X2 be linear operators on X1 and X2, respectively.

Then the 2-tensor product of T and S on X1

2
⊙X2, denoted by T

2
⊙S, is the linear operator

T
2
⊙ S : X1

2
⊙X2 −→ X1

2
⊙X2



M. Luis, F. Osmin, S. Arley / Eur. J. Pure Appl. Math, 18 (4) (2025), 6613 12 of 17

defined by (
T

2
⊙ S

)( n∑
i=1

xi
2
⊙ yi

)
:=

n∑
i=1

(
Txi

) 2
⊙
(
Syi
)
,

for each
∑n

i=1 xi
2
⊙ yi ∈ X1

2
⊙X2.

Remark 4. Let (X1, ⟨·, ·|·⟩1) and (X2, ⟨·, ·|·⟩2) be spaces with a generalized 2-inner product.

We denote the tensor product (I1
2
⊙ I2) on X1

2
⊙X2 by I⊙; this is the identity mapping on

the vector space X1

2
⊙X2. Indeed, for every ξ =

∑n
i=1 xi

2
⊙ yi ∈ X1

2
⊙X2,

I⊙(ξ) = (I1
2
⊙ I2)

( n∑
i=1

xi
2
⊙ yi

)
=

n∑
i=1

I1xi
2
⊙ I2yi =

n∑
i=1

xi
2
⊙ yi = ξ.

Note also that ∥I⊙∥ 2
⊙
= 1.

Proposition 8. Let (X1, ⟨·, · | ·⟩1) and (X2, ⟨·, · | ·⟩2) be spaces with a generalized 2-inner
product, let T1, S1 : X1 → X1, T2, S2 : X2 → X2 be linear operators, and let α, β ∈ C. Then
the following identities hold:

i) T1S1

2
⊙ T2S2 =

(
T1

2
⊙ T2

)
◦
(
S1

2
⊙ S2

)
;

ii) αβ (T1

2
⊙ T2) = (αT1)

2
⊙ (βT2);

iii) T1

2
⊙ (T2 + S2) = (T1

2
⊙ T2) + (T1

2
⊙ S2);

iv) (T1 + S1)
2
⊙ T2 = (T1

2
⊙ T2) + (S1

2
⊙ T2);

v) (T1 + S1)
2
⊙ (T2 + S2) = T1

2
⊙ T2 + T1

2
⊙ S2 + S1

2
⊙ T2 + S1

2
⊙ S2;

vi) T1

2
⊙ T2 is invertible if and only if both T1 and T2 are invertible, and in that case

(T1

2
⊙ T2)

−1 = T−1
1

2
⊙ T−1

2 .

Proof. The proofs follow directly by applying each operator definition to a simple tensor

sum ξ =
∑n

i=1 xi
2
⊙ yi and grouping like terms, using associativity and distributivity of

sums and scalars, as well as the definition of the tensor-product operator. The invertibility
statement in (vi) uses the fact that a tensor-product of invertible operators is itself
invertible, with inverse given by the tensor product of the individual inverses.

Example 3. If we consider (C2, ⟨·, ·|·⟩C2) and (C, ⟨·, ·|·⟩C) as in Example 2, then the
following mappings are linear operators

T : C2 → C2, T (a, b) = (0, a), (a, b) ∈ C2.



M. Luis, F. Osmin, S. Arley / Eur. J. Pure Appl. Math, 18 (4) (2025), 6613 13 of 17

S : C → C, S(c) = ic, c ∈ C,

and consequently, for each
∑n

k=1(ak, bk)
2
⊙ ck ∈ C2

2
⊙C the map T

2
⊙S : C2

2
⊙C → C2

2
⊙C

is given by

(
T

2
⊙ S

)( n∑
k=1

(ak, bk)
2
⊙ ck

)
=

n∑
k=1

T (ak, bk)
2
⊙ Sck =

n∑
i=1

(0, ak)
2
⊙ ick = i

n∑
i=1

(0, ak)
2
⊙ ck.

Now, for all ((x1, x2), d), ((y1, y2), e) ∈ C2 × C we have

(
i

n∑
i=1

(0, ak)
2
⊙ ck

)
(((x1, x2), d), ((y1, y2), e)) = i

n∑
k=1

⟨(0, ak), (x1, x2)|(y1, y2)⟩C2⟨ck, d|e⟩C

= i
n∑

k=1

akckx2d|y2e|2 = ix2d|y2e|2
n∑

k=1

akck

= ix2d|y2e|2⟨a, c̄⟩Cn

where a = (a1, · · · , an), c̄ = (c̄1, · · · , c̄n) ∈ Cn and ⟨·, ·⟩Cn denotes the classical inner
product on Cn.

Remark 5. Given two spaces with classical inner product (X1, ⟨·, ·⟩X1) and (X2, ⟨·, ·⟩X2),
and x ∈ X1, y ∈ X2, we know that x⊙y is defined as the mapping x⊙y : X1×X2 → C, given
by (x⊙ y)(r, s) = ⟨x, r⟩X⟨y, s⟩Y for all (r, s) ∈ X1 × X2 (see [11]). Now, we can consider
the application x⊙̃y : (X1 × X2) × (X1 × X2) → C, defined by (x⊙̃y)((x1, y1), (x2, y2)) :=
(x⊙ y)(x1, y1). Thus we obtain the following complex vector spaces

X1 ⊙X2 =

{
n∑

i=1

αi(xi ⊙ yi) : αi ∈ C, xi ∈ X1, yi ∈ X2, n ∈ N

}

X1⊙̃X2 =


m∑
j=1

βj(xj⊙̃yj) : βj ∈ C, xj ∈ X1, yj ∈ X2,m ∈ N


whose relationship is expressed in the following proposition:

Proposition 9. Let (X1, ⟨·, ·⟩1) and (X2, ⟨·, ·⟩2) spaces with a classical inner product. Then
X1 ⊙X2

∼= X1⊙̃X2.

Proof. Consider the linear transformation φ : X1 ⊙X2 → X1⊙̃X2 given by

φ(
n∑

i=1

xi ⊙ yi) =
n∑

i=1

xi⊙̃yi

note that clearly φ is bijective.



M. Luis, F. Osmin, S. Arley / Eur. J. Pure Appl. Math, 18 (4) (2025), 6613 14 of 17

Theorem 6. Let (X1, ⟨·, ·⟩1) and (X2, ⟨·, ·⟩2) spaces with a classical inner product. Then,

X1

2
⊙ X2 is a vector subspace of X1⊙̃X2, where the vector space X1

2
⊙ X2 is defined as in

the Definition 5.

Proof. It is enough to see that X1

2
⊙ X2 ⊆ X1⊙̃X2. In fact, whether ξ ∈ X1

2
⊙ X2 then

there is a natural number n, and there are {αi}ni=1 ⊆ C, {xi}ni=1 ⊆ X1, {yi}ni=1 ⊆ X2 such

that ξ =
∑n

i=1 αi(xi
2
⊙ yi). However, for all (x, y) ∈ X1 × X2 and all (r, s) ∈ X1 × X2 with

∥r∥X = 1, ∥s∥Y = 1 it is necessary to,

ξ((x, y), (r, s)) = (
n∑

i=1

αi(xi
2
⊙ yi))((x, y), (r, s)) =

n∑
i=1

(αi(xi
2
⊙ yi))((x, y), (r, s))

=
n∑

i=1

αi⟨xi, x|r⟩X⟨yi, y|s⟩Y =
n∑

i=1

αi⟨xi, x⟩X∥r∥2X⟨yi, y⟩Y ∥s∥2Y

=
n∑

i=1

αi⟨xi, x⟩X⟨yi, y⟩Y =
n∑

i=1

αi(xi⊙̃yi)((x, y), (r, s))

= (
n∑

i=1

αi(xi⊙̃yi))((x, y), (r, s))

Then, ξ =
∑n

i=1 αi(xi⊙̃yi) ∈ X1⊙̃X2. Therefore ξ ∈ X1⊙̃X2.

Proposition 10. Let (X1, ⟨·, · | ·⟩1) and (X2, ⟨·, · | ·⟩2) be spaces with a generalized 2-inner

product, if T1 ∈
2
B(X1), T2 ∈

2
B(X2) then T1

2
⊙ T2 ∈

2
B(X1

2
⊙X2).

Proof. Let ξ1 =
∑n

i=1 xi
2
⊙ yi, ξ2 =

∑m
k=1 rk

2
⊙ sk ∈ X1

2
⊙ X2 with ∥ξ1, ξ2∥ 2

⊙
= 1. Thus,

we have

∥∥∥∥(T1

2
⊙ T2)(ξ1), ξ2

∥∥∥∥22
⊙
=

〈
(T1

2
⊙ T2)(ξ1), (T1

2
⊙ T2)(ξ1)|ξ2

〉
2
⊙

=

〈
n∑

i=1

T1(xi)
2
⊙ T2(yi),

n∑
j=1

T1(xj)
2
⊙ T2(yj)|

m∑
k=1

rk
2
⊙ sk

〉
2
⊙

=

n∑
i=1

n∑
j=1

m∑
k=1

δi,j ⟨T1(xi), T1(xj)|rk⟩1 ⟨T2(yi), T2(yj)|sk⟩2

=
n∑

i=1

m∑
k=1

⟨T1(xi), T1(xj)|rk⟩1 ⟨T2(yi), T2(yj)|sk⟩2

=
n∑

i=1

m∑
k=1

∥T1(xi), rk∥21 ∥T2(yi), sk∥22



M. Luis, F. Osmin, S. Arley / Eur. J. Pure Appl. Math, 18 (4) (2025), 6613 15 of 17

≤
n∑

i=1

m∑
k=1

∥T1∥2 ∥xi, rk∥21 ∥T2∥2 ∥yi, sk∥22

= ∥T1∥2 ∥T2∥2
n∑

i=1

m∑
k=1

∥xi, rk∥21 ∥yi, sk∥
2
2

= ∥T1∥2 ∥T2∥2 ∥ξ1, ξ2∥22
⊙
= ∥T1∥2 ∥T2∥2

Therefore

∥∥∥∥(T1

2
⊙ T2)(ξ1), ξ2

∥∥∥∥ 2
⊙
≤ ∥T1∥ ∥T2∥ .

Proposition 11. Let (X1, ⟨·, ·|·⟩X1), (X2, ⟨·, ·|·⟩X2), (Y1, ⟨·, ·|·⟩Y1), (Y2, ⟨·, ·|·⟩Y2) spaces with
generalized 2-inner product and T1 : X1 → X2 and T2 : Y1 → Y2 linear operators, then

T1 × T2 : X1 × Y1 → X2 × Y2

(x1, y1) → (T1 × T2)(x1, y1) = T1(x1)⊙ T2(y2)

is a bilinear operator.

Proof. In fact, note that for all x1, x
′
1 ∈ X1 and all y1, y

′
1 ∈ Y1 is met

(T1 × T2)(αx1 + x′1, y1) = T1(αx1 + x′1)⊙ T2(y1) = (αT1(x1) + T1(x
′
1))⊙ T2(y1)

= (αT1(x1))⊙ T2(y1) + T1(x
′
1)⊙ T2(y1)

= α(T1(x1)⊙ T2(y1)) + T1(x
′
1)⊙ T2(y1)

= α(T1 × T2)(x1, y1) + (T1 × T2)(x
′
1, y1)

(T1 × T2)(x1, y1 + y′1) = T1(x1)⊙ T2(y1 + y′1) = T1(x1)⊙ ((T2(αy1) + T2(y
′
1))

= T1(x1)⊙ (αT2(y1)) + T1(x1)⊙ T2(y
′
1)

= α(T1(x1)⊙ T2(y1)) + T1(x1)⊙ T2(y
′
1)

= α(T1 × T2)(x1, y1) + (T1 × T2)(x1, y
′
1)

5. Conclusions

In this research, we introduced the notion of tensor product of elements of spaces
equipped with a generalized 2-inner product, calling it the 2-tensor product (see Definition
5), which turns out to be a bilinear mapping in the sense of Proposition 5, as in the classic
case. We also define the algebraic tensor product of spaces endowed with a generalized
2-inner product and prove that it satisfies all fundamental properties (see Proposition
5). Furthermore, we define a generalized 2-inner product on the algebraic tensor product
(see Definition 6) and induced a 2-norm, which we call the induced 2-tensor norm (see



M. Luis, F. Osmin, S. Arley / Eur. J. Pure Appl. Math, 18 (4) (2025), 6613 16 of 17

Theorem 1). We also established the notion of the 2-tensor product of linear operators
on spaces with a generalized 2-inner product (see Definition 9), and demonstrated that
several well-known identities for operators in inner product spaces remain valid in this
new structure (see Proposition 8). Finally, the notions established in this work have been
illustrated with examples (see Example 2 and Example 3). In this context, other theories
can be developed, such as frame theory, soft set theory, the theory of functions of bounded
variation, the study of the numerical range of operators, and others.

Conflict of interest

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

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	Introduction
	Preliminaries
	Main Results
	2-Tensor Product
	Algebraic Tensor Product

	Tensor Product of Linear Operators
	Conclusions