EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 3, 2024, 1937-1947
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

On Inner Products Derived From the Standard n-Inner
Product on an Inner Product Space

Adam Adam1∗, Steven Rante2, Hendra Gunawan2

1 Department of Mathematics, Institut Teknologi Kalimantan, Balikpapan 76127, Indonesia
2 Analysis and Geometry Group, Faculty of Mathematics and Natural Sciences,
Bandung Institute of Technology, Bandung 40132, Indonesia

Abstract. In this paper, we study relations between inner products derived from the standard
n-inner product defined on an inner product space. In particular, we are interested in knowing
when orthogonality with respect to the original inner product is preserved by the derived inner
product.

2020 Mathematics Subject Classifications: 46C50, 46B20, 46C05

Key Words and Phrases: Inner products, n-inner products, orthogonality, orthogonal set,
orthonormal basis.

1. Introduction

Let X be a real vector space of dimension d ≥ n and let ⟨·, ·|·, . . . , ·⟩ : Xn+1 → R be a
function such that for every x0, x1, . . . , xn, xn+1 ∈ X and α ∈ R we have

(I1) ⟨x1, x1|x2, . . . , xn⟩ ≥ 0 and ⟨x1, x1|x2, . . . , xn⟩ = 0 if and only if x1, x2, . . . , xn are
linearly dependent;

(I2) ⟨x1, x1|x2, . . . , xn⟩ = ⟨xi1 , xi1 |xi2 , . . . , xin⟩ for any permutation; {i1, i2, . . . , in} of
(1, . . . , n);

(I3) ⟨x0, x1|x2, . . . , xn⟩ = ⟨x1, x0|x2, . . . , xn⟩;

(I4) ⟨αx0, x1|x2, . . . , xn⟩ = α⟨x0, x1|x2, . . . , xn⟩;

(I5) ⟨x0 + xn+1, x1|x2, . . . , xn⟩ = ⟨x0, x1|x2, . . . , xn⟩+ ⟨xn+1, x1|x2, . . . , xn⟩.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i3.5263

Email addresses: adam@lecturer.itk.ac.id (Adam Adam),
stevengrante@gmail.com (Steven Rante), hgunawan@math.itb.ac.id (Hendra Gunawan)

https://www.ejpam.com 1937 © 2024 EJPAM All rights reserved.



A. Adam, S. Rante, H. Gunawan / Eur. J. Pure Appl. Math, 17 (3) (2024), 1937-1947 1938

The function ⟨·, ·|·, . . . , ·⟩ is called an n-inner product introduced by Misiak in 1989
[14]. Here, the pair (X, ⟨·, ·|·, . . . , ·⟩) is called an n-inner product space. If (X, ⟨·, ·⟩) is a
real inner product space of dimension d ≥ n, we can define the standard n-inner product
by

⟨x0, x1|x2, . . . , xn⟩ =

∣∣∣∣∣∣∣∣∣
⟨x0, x1⟩ ⟨x0, x2⟩ . . . ⟨x0, xn⟩
⟨x2, x1⟩ ⟨x2, x2⟩ . . . ⟨x2, xn⟩

...
...

. . .
...

⟨xn, x1⟩ ⟨xn, x2⟩ . . . ⟨xn, xn⟩

∣∣∣∣∣∣∣∣∣.
So, a real inner product space of dimension d ≥ n with the n-inner product defined above
is an example of an n-inner product space, which we can call a standard n-inner product
space.

Next, from n-inner product space we can derive n-norm, defined by

∥x1, x2, . . . , xn∥ = ⟨x1, x1|x2, . . . , xn⟩
1
2 .

Furthermore, an n-norm on X is a function ∥·, . . . , ·∥ : Xn → R such that for every
x0, x1, . . . , xn, xn+1 ∈ X and α ∈ R, the function satisfying the following properties:

(N1) ∥x1, x2, . . . , xn∥ ≥ 0 and ∥x1, x2, . . . , xn∥ = 0 if and only if x1, x2, . . . , xn are linearly
dependent;

(N2) ∥x1, x2, . . . , xn∥ is invariant under permutation;

(N3) ∥αx1, x2, . . . , xn∥ = |α| ∥x1, x2, . . . , xn∥;

(N4) ∥x0 + x1, x2, . . . , xn∥ ≤ ∥x0, x2, . . . , xn∥+ ∥x1, x2, . . . , xn∥.

Geometrically, ∥x1, x2, . . . , xn∥ represents the generalized volume of an n-dimensional
parallelepiped spanned by x1, x2, . . . , xn. Then,

⟨x0, x1|x2, . . . , xn⟩
∥x0, x2, . . . , xn∥ ∥x1, x2, . . . , xn∥

is the cosine of the angle between two parallelepipeds spanned by x0, x2, . . . , xn and
x1, x2, . . . , xn. See [9, 10, 15] for more properties of n-inner products. The related results
may also be found in [3–6, 13, 16, 17].

Historically, numerous authors have introduced and developed several concepts of or-
thogonality in 2-normed spaces and 2-inner product spaces [1, 2, 7, 11, 12, 15]. Just as the
concepts of orthogonality in normed spaces draw inspiration from those in inner product
spaces, the notions of orthogonality in 2-normed spaces are similarly linked to those in
2-inner product spaces. In [11], it is shown that the standard definition of orthogonality
in a 2-inner product space (X, ⟨·, ·|·⟩) with dim(X) ≥ 3, is as follows:



A. Adam, S. Rante, H. Gunawan / Eur. J. Pure Appl. Math, 17 (3) (2024), 1937-1947 1939

Definition 1 (G-orthogonality in 2-inner product spaces). Let (X, ⟨·, ·|·⟩) be a 2-inner
product spaces. x1 is G-orthogonal to x2 if and only if there exists a subspace V of X with
codim(V ) = 1 such that ⟨x1, x2|x⟩ = 0 for all x ∈ V (Denoted by x1⊥Gx2).

We can say this definition is standard because when X is a standard 2-inner product
space, we have x1⊥x2 if and only if x1⊥Gx2. The definition of G-orthogonality provided
above represents an enhancement over the definition proposed by Cho and Kim [2] and
Godini [7] as demonstrated in [11]. On the other hand, Cho and Kim’s concept of or-
thogonality and Godini can be seen as a development of Khan and Siddiqui’s concept of
orthogonality [12]. Furthermore, one can define the notion of G-orthogonality in n-inner
product spaces as follows:

Definition 2 (G-orthogonality in n-inner product spaces). Let (X, ⟨·, ·|·, . . . , ·⟩) be an n-
inner product spaces with dim(X) ≥ n + 1. x1 is G-orthogonal to x2 if and only if there
exists a subspace V of X with codim(V ) = 1 such that ⟨x1, x2|x3, . . . , xn+1⟩ = 0 for all
x3, . . . , xn+1 ∈ V (Denoted by x1⊥Gx2).

With this definition, in a standard n-inner product space, G-orthogonality is also
equivalent to the usual orthogonality (with respect to the inner product). In other words,
x1⊥x2 if and only if x1⊥Gx2.

We can see that the definition requires the assumption that the dimension of X is
greater than n. In [11], it is shown that if we define G-orthogonality in the standard
2-inner product space X of dimension 2, any pair of linearly independent vectors becomes
G-orthogonal. Similarly, if we define G-orthogonality for a standard n-inner product space
X of dimension n, any pair of linearly independent vectors also becomes G-orthogonal.
Indeed, this fact is undesirable. It is necessary to adopt a different approach to establish
orthogonality in n-inner product spaces of dimension n in a general sense.

Meanwhile, note that if (X, ⟨·, ·|·, . . . , ·⟩) is an arbitrary n-inner product space and
A = {a1, a2, . . . , an} is a set of n linearly independent vectors in X, then one may observe
that

⟨x, y⟩A :=
∑

{i2,...in}⊂{1,2,...,n}

⟨x, y|ai2 , . . . , ain⟩

defines an inner product on X. There are n terms in the above sum, as there are n
subsets of {1, 2, . . . , n} consisting of n− 1 elements. If dimX = d < ∞, we can also define
an inner product by the above formula using a set of d linearly independent vectors in X
(see [8]).

Thus, starting from an inner product, we can define the standard n-inner product, and
then from the n-inner product, we can derive a new inner product. It is then interesting
to investigate how the new inner product derived from standard n-inner product relates
to the original inner product on (X, ⟨·, ·⟩). In particular, we would like to know whether



A. Adam, S. Rante, H. Gunawan / Eur. J. Pure Appl. Math, 17 (3) (2024), 1937-1947 1940

or not the new inner product preserves orthogonality. This generally depends on the set
A that we choose in the definition of the new inner product. In the next sections, we
present necessary and sufficient conditions for the set A to give the positive answer. With
this approach, we have an alternative way to establish the orthogonality of two vectors
in arbitrary n-inner product spaces because we can define the inner product on n-inner
product spaces.

2. The n-dimensional case

Let (X, ⟨·, ·|·, . . . , ·⟩) be a standard n-inner product space. As indicated in [10] and
[11], the n-dimensional case is special. So we shall first pay attention to the case where
dimX = n. Our results are the following theorems.

Theorem 1. Let A = {a1, a2, . . . , an} ⊂ X be an orthogonal set with ∥ai∥ = α > 0 for all
i = 1, 2, . . . , n. Then ⟨x, y⟩A = 0 if and only if ⟨x, y⟩ = 0 for all x, y ∈ X.

Proof. For any subset {i2, . . . , in} ⊂ {1, 2, . . . , n}, we have

⟨x, y|ai2 , ai3 , . . . , ain⟩ =

∣∣∣∣∣∣∣∣∣∣∣

⟨x, y⟩ ⟨x, ai2⟩ . . . ⟨x, ain⟩
⟨ai2 , y⟩ ⟨ai2 , ai2⟩ . . . ⟨ai2 , ain⟩

...
...

. . .
...

⟨ain , y⟩ ⟨ain , ai2⟩ . . . ⟨ain , ain⟩

∣∣∣∣∣∣∣∣∣∣∣
n×n

= ⟨x, y⟩

∣∣∣∣∣∣∣∣∣
∥ai2∥

2 0 . . . 0

0 ∥ai3∥
2 . . . 0

...
...

. . .
...

0 0 . . . ∥ain∥
2

∣∣∣∣∣∣∣∣∣
(n−1)×(n−1)

− ⟨x, ai2⟩

∣∣∣∣∣∣∣∣∣
⟨ai2 , y⟩ 0 . . . 0

⟨ai3 , y⟩ ∥ai3∥
2 . . . 0

...
...

. . .
...

⟨ain , y⟩ 0 . . . ∥ain∥
2

∣∣∣∣∣∣∣∣∣
(n−1)×(n−1)

+ · · ·+ (−1)n−1⟨x, ain⟩

∣∣∣∣∣∣∣∣∣
⟨ai2 , y⟩ ∥ai2∥

2 . . . 0
...

...
. . .

...

⟨ain−1 , y⟩ 0 . . .
∥∥ain−1

∥∥2
⟨ain , y⟩ 0 . . . 0

∣∣∣∣∣∣∣∣∣
(n−1)×(n−1)

= ⟨x, y⟩
n∏

j=2

∥∥aij∥∥2 − ⟨x, ai2⟩⟨ai2 , y⟩
n∏

j=3

∥∥aij∥∥2 − · · · − ⟨x, ain⟩⟨ain , y⟩
n−1∏
j=2

∥∥aij∥∥2
=

[
⟨x, y⟩ − ⟨x, ai2⟩⟨ai2 , y⟩

∥ai2∥
2 − ⟨x, ai3⟩⟨ai3 , y⟩

∥ai3∥
2 − · · · − ⟨x, ain⟩⟨ain , y⟩

∥ain∥
2

]
n∏

j=2

∥∥aij∥∥2 .



A. Adam, S. Rante, H. Gunawan / Eur. J. Pure Appl. Math, 17 (3) (2024), 1937-1947 1941

Let {1, 2, . . . n}\{i2, . . . in} = {i1}. By Parseval’s identity, we have ⟨x, y⟩ =
n∑

j=1

⟨x, aij ⟩⟨aij , y⟩∥∥aij∥∥2
(since {i1, i2, . . . , in} = {1, 2, . . . , n} as sets). Hence it follows that

⟨x, y|ai2 , ai3 , . . . , ain⟩ =
⟨x, ai1⟩⟨ai1 , y⟩

∥ai1∥
2

n∏
j=2

∥∥aij∥∥2 = ⟨x, ai1⟩⟨ai1 , y⟩
∥ai1∥

4

n∏
i=1

∥ai∥2 .

Summing the above expressions for i1 = 1, 2, . . . , n, we get

⟨x, y⟩A =
∑

{i2,...in}⊂{1,2,...,n}

⟨x, y|ai2 , . . . , ain⟩

=

[
⟨x, a1⟩⟨a1, y⟩

∥a1∥4
+

⟨x, a2⟩⟨a2, y⟩
∥a2∥4

+ · · ·+ ⟨x, an⟩⟨an, y⟩
∥an∥4

]
n∏

i=1

∥ai∥2 .

However, we are assuming that ∥ai∥ = α for all i = 1, 2, . . . , n and so we obtain

⟨x, y⟩A =

[
⟨x, a1⟩⟨a1, y⟩

α4
+

⟨x, a2⟩⟨a2, y⟩
α4

+ · · ·+ ⟨x, an⟩⟨an, y⟩
α4

]
n∏

i=1

α2

=
⟨x, y⟩
α2

α2n = α2(n−1)⟨x, y⟩.

Since α ̸= 0, we conclude that ⟨x, y⟩A = 0 if and only if ⟨x, y⟩ = 0, which proves the
theorem.

Theorem 2. Let A = {b1, b2, . . . , bn} be a set of n linearly independent vectors in X.
Then ⟨x, y⟩A = ⟨x, y⟩ if and only if A is an orthonormal basis for X.

Proof. The sufficient part follows immediately from the previous theorem. For the
necessary part, suppose that ⟨x, y⟩A = ⟨x, y⟩ for all x, y ∈ X. To prove that A is an
orthonormal basis for X, let us first compute (bi, bj) for i ̸= j. We have

⟨bi, bj⟩ = ⟨bi, bj⟩A =
∑

{i2,...in}⊂{1,2,...,n}

⟨bi, bj |bi2 , . . . , bin⟩.

For any {i2, . . . , in} ⊂ {1, 2, . . . , n}, observe that bi ∈ {bi2 , . . . , bin} or bj ∈ {bi2 , . . . , bin},
because {bi2 , . . . , bin} consists of n−1 elements of A. Consequently , ⟨bi, bj |bi2 , . . . , bin⟩ = 0,
because two rows or two columns in the determinant will be identical. Since this is true
for any {i2, . . . , in} ⊂ {1, 2, . . . , n}, we conclude that

⟨bi, bj⟩ =
∑

{i2,...in}⊂{1,2,...,n}

⟨bi, bj |bi2 , . . . , bin⟩ = 0.

Let us now compute ⟨bi, bi⟩ for i = 1, 2, . . . , n. Notice that if bi ∈ {bi2 , . . . , bin}, then we
have ⟨bi, bi|bi2 , . . . , bin⟩ = 0. Meanwhile, if bi ̸∈ {bi2 , . . . , bin} then —by the properties of
the standard n−product— we have



A. Adam, S. Rante, H. Gunawan / Eur. J. Pure Appl. Math, 17 (3) (2024), 1937-1947 1942

⟨bi, bi|bi2 , . . . , bin⟩ = ⟨b1, b1|b2, . . . , bn⟩ =

∣∣∣∣∣∣∣∣∣
⟨b1, b1⟩ ⟨b1, b2⟩ . . . ⟨b1, bn⟩
⟨b2, b1⟩ ⟨b2, b2⟩ . . . ⟨b2, bn⟩

...
...

. . .
...

⟨bn, b1⟩ ⟨bn, b2⟩ . . . ⟨bn, bn⟩

∣∣∣∣∣∣∣∣∣ =
n∏

j=1

∥bj∥2 .

Hence, we obtain

⟨bi, bi⟩A =
∑

{i2,...in}⊂{1,2,...,n}

⟨bi, bi|bi2 , . . . , bin⟩ =
n∏

j=1

∥bj∥2 .

By our hypothesis, ∥bi∥2 = ∥bi∥2A =
n∏

j=1

∥bj∥2. This holds only if ∥bi∥ = 1 for all

i = 1, 2, . . . , n.
To sum up, we have proved that ⟨x, y⟩A = ⟨x, y⟩ if and only if A is an orthonormal

basis for X.

3. The higher dimensional case

Let us now consider the case where n+1 ≤ d = dim X < ∞. Let A := {a1, a2, . . . , ad}
be a set of linearly independent vectors in X. (What happens if we use only n vectors will
be discussed later, together with the case where d = ∞.) We define the following inner
product on X:

⟨x, y⟩A :=
∑

{i2,...in}⊂{1,2,...,n}

⟨x, y|ai2 , . . . , ain⟩.

Note that there are

(
d

n− 1

)
terms in the above sum. Analogous to Theorem 2.1, we

have the following theorem.

Theorem 3. Let A = {a1, a2, . . . , ad} ⊂ X be an orthogonal set with ∥ai∥ = α > 0 for all
i = 1, 2, . . . , d. Then ⟨x, y⟩A = 0 if and only if ⟨x, y⟩ = 0 for all x, y ∈ X.

Proof. Let Id := {1, 2, . . . , d}. For any I0 := {i2, . . . , in} ⊂ Id, let I1 := Id \ I0. Then,
we have

⟨x, y|ai2 , . . . ain⟩ =
∑
i∈I1

⟨x, ai⟩⟨ai, y⟩
∥ai∥2

∏
i∈I0

∥ai∥2

=

[
⟨x, y⟩ −

∑
i∈I0

⟨x, ai⟩⟨ai, y⟩
∥ai∥2

] ∏
i∈I0

∥ai∥2 .



A. Adam, S. Rante, H. Gunawan / Eur. J. Pure Appl. Math, 17 (3) (2024), 1937-1947 1943

Summing over all subsets I0 ⊂ Id and using the assumption that ∥ai∥ = α for all
i = 1, 2, . . . n, we obtain

⟨x, y⟩A =
∑
I0⊂Id

[
⟨x, y⟩ − ⟨x, ai⟩⟨ai, y⟩

∥ai∥2

]
α2(n−1)

=
∑
I0⊂Id

α2(n−1)⟨x, y⟩ −
∑
I0⊂Id

∑
i∈I0

α2(n−1) ⟨x, ai⟩⟨ai, y⟩
α2

.

The first sum on the right hand side is equal to

(
d

n− 1

)
α2(n−1)⟨x, y⟩. For the second

sum, we observe that each expression ⟨x, ai⟩⟨ai, y⟩ occurs precisely
(
d− 1
n− 2

)
times for all

i ∈ Id.

Hence, by Parseval’s identity, the second sum is equal

(
d− 1
n− 2

)
α2(n−1)⟨x, y⟩. Therefore

we get

⟨x, y⟩A =

[(
d

n− 1

)
−
(
d− 1
n− 2

)]
α2(n−1)⟨x, y⟩ =

(
d− 1
n− 1

)
α2(n−1)⟨x, y⟩,

which gives us the desired conclusion.

Corollary 1. If A = {a1, a2, . . . , ad} is an orthonormal basis for X, then ⟨x, y⟩A =(
d− 1
n− 1

)
⟨x, y⟩ for all x, y ∈ X.

Remark 1. The converse of the above corollary does not hold. To give an example, let
d = dimX = 3 and n = 2. Let A = {a1, a2, a3} be linearly independent set in X. Suppose
that ⟨x, y⟩A = 2⟨x, y⟩ for all x, y ∈ X. We would like to check whether we have ∥ai∥ = 1
for i = 1, 2, 3 and ⟨ai, aj⟩ = 0 for i ̸= j. Notice that

⟨x, y⟩A =

3∑
i=1

⟨x, y|ai⟩ =
3∑

i=1

∣∣∣∣ ⟨x, y⟩ ⟨x, ai⟩
⟨ai, y⟩ ⟨ai, ai⟩

∣∣∣∣
for all x, y ∈ X. From the hypothesis, we have ⟨ai, aj⟩A = 2⟨ai, aj⟩ for i, j = 1, 2, 3, which
may be rewritten as

∥a1∥2 ∥a2∥2 − ⟨a1, a2⟩2 + ∥a1∥2 ∥a3∥2 − ⟨a1, a3⟩2 = 2 ∥a1∥2 ,
∥a1∥2 ∥a2∥2 − ⟨a1, a2⟩2 + ∥a2∥2 ∥a3∥2 − ⟨a2, a3⟩2 = 2 ∥a2∥2 ,
∥a1∥2 ∥a3∥2 − ⟨a1, a3⟩2 + ∥a2∥2 ∥a3∥2 − ⟨a2, a3⟩2 = 2 ∥a3∥2 ,

⟨a1, a2⟩ ∥a3∥2 − ⟨a1, a3⟩⟨a2, a3⟩ = 2⟨a1, a2⟩,
⟨a1, a3⟩ ∥a2∥2 − ⟨a1, a2⟩⟨a2, a3⟩ = 2⟨a1, a3⟩,
⟨a2, a3⟩ ∥a1∥2 − ⟨a1, a2⟩⟨a1, a3⟩ = 2⟨a2, a3⟩.



A. Adam, S. Rante, H. Gunawan / Eur. J. Pure Appl. Math, 17 (3) (2024), 1937-1947 1944

Let A := ∥a1∥ , B := ∥a2∥ , C := ∥a3∥ , D := ⟨a1, a2⟩, E := ⟨a1, a3⟩, F := ⟨a2, a3⟩. Then

A2B2 −D2 +A2C2 − E2 = 2A2,

A2B2 −D2 +B2C2 − F 2 = 2B2,

A2C2 − E2 +B2C2 − F 2 = 2C2,

DC2 − EF = 2D,

EB2 −DF = 2E,

FA2 −DE = 2F.

Observe that A = B = C = 1, D = E = F = 0 satisfy the above equations simultaneously.
We shall see that there are other possible solutions with D,E, F ̸= 0.

Multiplying both sides in the last three equations by D,E, and F (respectively) and
rearranging the terms, we obtain

D2 + E2 = A2B2 +A2C2 − 2A2, (1)

D2 + F 2 = A2B2 +B2C2 − 2B2, (2)

E2 + F 2 = A2C2 +B2C2 − 2C2, (3)

C2D2 − 2D2 = DEF, (4)

B2E2 − 2E2 = DEF, (5)

A2F 2 − 2F 2 = DEF. (6)

From (1), (2), and (3), we get

D2 = A2B2 −A2 −B2 + C2, (7)

E2 = A2C2 −A2 +B2 − C2, (8)

F 2 = B2C2 +A2 −B2 − C2. (9)

From (4), (5), and (6), we get

(C2 − 2)D2 = (B2 − 2)E2 = (A2 − 2)F 2 = DEF. (10)

Substituting (7), (8), and (9) into (10), we obtain

(B2 − C2)(A2 +B2 + C2 − 4) = 0,

(A2 − C2)(A2 +B2 + C2 − 4) = 0,

(A2 −B2)(A2 +B2 + C2 − 4) = 0.

Now one may check that A = B = C = 4
3 , D + E + F = −2

3 satisfy the above equations
simultaneously. This tells us that A is not necessarily an orthonormal basis.



A. Adam, S. Rante, H. Gunawan / Eur. J. Pure Appl. Math, 17 (3) (2024), 1937-1947 1945

We now come to the case where d = dim X = ∞. We assume that X is separable
and B := {ai : i = 1, 2, 3, . . . } is an orthogonal basis for X. Thus for all x, y ∈ X,

we have Parseval’s identity that
∞∑
i=1

⟨x, ai⟩⟨ai, y⟩
∥ai∥2

will converge to ⟨x, y⟩. Next, let A :=

{a1, a2, . . . , an}, where the vectors ai’s are the first n vectors in B. We define

⟨x, y⟩A :=
∑

{i2,...,in}⊂{1,2,...,n}

⟨x, y|ai2 , . . . , ain⟩

for all x, y ∈ X. Then we have the following theorem.

Theorem 4. For all x, y ∈ X, we have

⟨x, y⟩A =

[
n∑

i=1

⟨x, ai⟩⟨ai, y⟩
∥ai∥4

+ n
∞∑

i=n+1

⟨x, ai⟩⟨ai, y⟩
∥ai∥2

]
n∏

j=1

∥aj∥2 .

In particular, if ∥ai∥ = α for i = 1, 2, . . . , n then

⟨x, y⟩A = α2(n−1)

[
n∑

i=1

⟨x, ai⟩⟨ai, y⟩
α2

+ n
∞∑

i=n+1

⟨x, ai⟩⟨ai, y⟩
∥ai∥2

]
.

Proof. The proof is similar to the proof of Theorem 2.1 but this time we have

⟨x, y|ai2 , . . . , ain⟩ =

[
⟨x, ai1⟩⟨ai1 , y⟩

∥ai1∥
4 +

1

∥ai1∥
2

∞∑
i=n+1

⟨x, ai1⟩⟨ai1 , y⟩
∥ai∥2

]
n∏

j=1

∥aj∥2 ,

where {i1} = {1, 2, . . . , n} \ {i2, . . . , in}. Summing all these expression for all subsets
{i2, . . . , in} ⊂ {1, 2, . . . , n}, we obtain

⟨x, y⟩A =

[
n∑

i=1

⟨x, ai⟩⟨ai, y⟩
∥ai∥4

+

n∑
i=1

1

∥ai∥2
∞∑

i=n+1

⟨x, ai⟩⟨ai, y⟩
∥ai∥2

]
n∏

j=1

∥aj∥2 .

In particular, if ∥ai∥ = α for i = 1, 2, . . . , n, then we have

⟨x, y⟩A = α2(n−1)

[
n∑

i=1

⟨x, ai⟩⟨ai, y⟩
α2

+ n

∞∑
i=n+1

⟨x, ai⟩⟨ai, y⟩
∥ai∥2

]
as claimed.

Remark 2. Note that if ∥ai∥ = 1 for i = 1, 2, 3, . . . (that is, B is an orthonormal basis
for X), then the conclusion in the above theorem tells us that

⟨x, y⟩A =
n∑

i=1

⟨x, ai⟩⟨ai, y⟩+ n
∞∑

i=n+1

⟨x, ai⟩⟨ai, y⟩

for all x, y ∈ X.



REFERENCES 1946

Corollary 2. Suppose that ∥ai∥ = α for i = 1, 2, . . . , n. For every x, y ∈ X, let x :=
xA+x⊥A and y := yA+y⊥A where xA and yA are the orthogonal projections of x and y on span
A (respectively), and x⊥A and y⊥A are their complements (respectively). If ⟨xA, yA⟩A = 0
and ⟨x⊥A, y⊥A⟩ = 0, then ⟨x, y⟩ = 0. Conversely, if ⟨xA, yA⟩ = 0 and ⟨x⊥A, y⊥A⟩ = 0, then
⟨x, y⟩A = 0.

Acknowledgements

The first author is supported by LPPM Institut Teknologi Kalimantan. The third
author is supported by P3MI-ITB Program. The authors express their gratitude to the
editor and anonymous reviewers for their valuable comments that helped improve the
quality of this work.

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