INTERNATIONAL JOURNAL OF COMPUTING SCIENCE AND APPLIED MATHEMATICS.
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Construction of Cone 2-Norm Associated with S-Cone Inner Product Sadjidon.
Mahmud Yunus.
Sunarsini, and Lukman Hanafi AbstractAiThis paper is devoted to discussing an inner product in cone normed spaces and constructing S-cone inner products to define S-cone inner product spaces, especially in Ee2 -space.
Moreover, we also construct cone 2-norm spaces associated with S-cone inner product spaces.
Index TermsAiCone Normed Spaces.
S-cone inner product spaces.
Cone 2-Norm Spaces I NTRODUCTION HE study of 2-norm space continues to grow and learn.
among others, the study of 2-norms by associating its dual space that has been studied in .
, especially for the Ee2 -space and the inner product space, and also in .
by studying the cone normed space.
Therefore, taking into account in .
is developed a study of the cone 2-norm and some of its properties described in .
Therefore, concerning inner product space and in .
it has been developed and studied about the S-cone inner product space, then they obtained the construction and definition of S-cone inner product spaces, particularly for Ee2 -space.
They also describe its properties, and construct its cone 2-normed such that is obtained a definition of cone 2-normed associated with a S-cone inner product.
To construct the S-cone inner product space, particularly for Ee2 -space, we need and use the following definitions and Definition 1.
Let X be a real vector space.
A norm on X is a function Ou A Ou : X Ie R satisfying:
(N.
OuxOu Ou 0 for every x OO X and OuxOu = 0 if and only if x = 0.
(N.
OuxOu = || OuxOu for every x OO X and OO R.
(N.
Oux yOu = OuxOu OuyOu.
A vector space X equipped with a norm OuAOu, written as (X.
OuAO.
, is called normed space.
Definition 2.
Let X be a real vector space.
An inner product on X is a function A.
A : X y X Ie R that satisfies:
(I.
x, x Ou 0 for every x OO X.
and x, x = 0 if and only if x = 0.
(I.
x, y = y, x.
(I.
x, y = x, y for every x, y OO X and OO R.
(I.
x y, z = x, z y, z for every x, y, z OO X.
Sadjidon.
Yunus.
Sunarsini, and L.
Hanafi are with the Department of Mathematics.
Institut Teknologi Sepuluh Nopember.
Surabaya 60111.
Indonesia e-mail: sadjidon@matematika.
Manuscript received August 3, 2023.
accepted November 1, 2023.
A vector space X equipped with an inner product A, ]cdot, also written as (X.
A), is called inner product space.
Definition 3.
Let P be a subset of a Banach space E with zero element , then P is called cone if:
P is a closed non empty set, and P = {}.
If a and b are positive real numbers, then ax by OO P for every x, y OO P .
P O (OeP ) = {}.
Additionally, a cone P has a relation O and x O y if and only if y Oe x OO P and x O y if and only if x O y and x = y, while x O y means y Oe x OO int(P ) .
nterior of P ).
Furthermore, we assume that E is the Banach space and P is a cone in E.
Definition 4.
A cone normed space is an ordered pair (X.
OuAOuc ) where X is a linear space over R and OuAOuc : X Ie (E.
OuAO.
is a function satisfying (C.
OuxOuc O for every x OO X.
(C.
OuxOuc = if and only if x = 0.
(C.
OuxOuc = || OuxOuc for every x OO X and OO R.
(C.
Oux yOuc O OuxOuc OuyOuc for every x, y OO X.
Definition 5.
Let x be a d-dimensional real vektor space, where 2 O d < O.
A 2-norm on X is a function OuA.
AOu :
X y X Ie R satisfying (N.
Oux, yOu Ou 0 for ef=very x, y OO X.
and Oux, yOu = 0 if and only if x and y are linearly dependent.
(N.
Oux, yOu = Ouy, xOu for every x, y OO X.
(N.
Oux, yOu = || Oux, yOu for every x, y OO X and OO R.
(N.
Oux, y zOu O Oux, yOu Oux, zOu for every x, y, z OO X.
A vector space X equipped with a 2-norm, also written as (X.
OuA.
AO.
, is called 2-norm space.
For historical issues regarding inner product spaces and 2normed spaces, we refer to the existing references.
, .
, in which defined a standard norm:
Oux, yOu = x, x y, x x, y y, y = x, x y, y Oe x, y = OuxOu OuyOu Oe x, y .
As in .
, we define the 2-norm by associating its dual space with x, z:
x, y y, z Oux, yOu = sup : z, w OO Ee2 .
OuzOu.
OuwOu O 1 .
x, w y, w Geometrically, the 2-norm is the area spanned by two vectors.
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Definition 6.
Let X be a 2-normed space, and (E.
OuAO.
be a Banach space, and P OC E be a cone, then cone 2-norm on X is a function OuA.
AOuC : X y X Ie (E.
OuAO.
satisfying the following properties:
(CN.
Oux, yOuC O for every x, y OO X.
and Oux, yOuC = if and only if x and y are linearly dependent.
(CN.
Oux, yOuC = Ouy, xOuC for every x, y OO X.
(CN.
Oux, yOuC = || Oux, yOuC for every x, y OO X and OO (CN.
Oux, y zOuC O Oux, yOuC Oux, zOuC for every x, y, z OO A 2-normed space X equipped with cone 2-norm, written as (X.
OuA.
AOuC ), is called cone 2-normed space.
(IC.
x, yC = k=1 ek x, y = k=1 ek y, x = y, xC y, xC for every x, y OO Ee2 .
Therefore x, yC = y, xC .
(IC.
x, yC = k=1 ek x, y = k=1 ek x, y = x, yC for every x, y OO Ee2 and OO R.
(IC.
For every x, y, z OO Ee2 by triangle inequality of the inner product, we have x y, zC = It is straightforward to verify that if P OC Rn for nonnegative R, then P is a cone.
As stated in P .
, a function OuAOuC : Ee2 Ie (Rn .
OuAO.
defined by OuxOuC = k=1 ek OuxOuEe2 is a cone normed space.
Multiplication on a cone norm is defined as follows:
ek OuxOuEe2 ek x y, z ek .
, z y, .
ek x, z II.
R ESULTS AND D ISCUSSION
OuxOuC OuxOuC = (OuxOuC ) = Let P be a subset of Banach space E and P is a cone, then we define P = P O (OeP ), and P is called S-cone.
Thus, from the description of the inner product space and the meaning of S-cone, we can construct and define a S-cone inner product space as in the following definition.
Definition 7.
A S-cone inner product space is an order pair (X.
AC ) where X a linear space overR with P is a S-cone inner product space and A.
AC : X y X Ie (E.
OuAO.
is a function satisfying:
(IC.
x, xC O for every x OO X.
and x, xC = if and only if x = 0.
(IC.
x, yC = y, xC for every x, y OO X.
(IC.
x, yC = x, yC for every x, y OO X and OO R.
(IC.
x y, zC O x, zC y, zC for every x, y, z OO X.
If X is a real vector space, then y, xC = y, xC = x, yC .
It is easy to show that Ee2 -space with the standard inner product is a Banach space, and its S-cone inner product is given as follows.
Theorem 1.
Let (Ee2 .
A) be an inner product space with P is a S-cone, and we define a function A.
AC : Ee2 y Ee2 Ie P (Rn .
OuAO.
by x, yC = k=1 ek x, y , .
then A.
AC is a S-cone inner product for Ee2 -space.
Proof.
We will show that A.
AC in .
satisfies the following (IC.
Since x, x Ou 0, then x, xC = k=1 ek x, xC O P for every x OO Ee2 .
Furthermore, x, xC = k=1 ek x, x = if and only if x, x = if and only if x = 0.
ek y, z = x, zc y, zC .
Therefore, we can conclude that the function A.
AC in .
is a S-cone inner product for Ee2 -space.
n In an inner product space, vectors x and x are orthogonal if and only if x, y = 0.
We define orthogonality in S-cone inner product spaces analogously to those in the inner product space, which is given by the following theorems.
Theorem 2.
Let (Ee2 .
A) be an inner product space with P is a S-cone, and if we define a S-cone Pninner product A.
AC :
Ee2 y Ee2 Ie (R.
OuAO.
by x, yC = k=1 ek x, y, then two vectors x and y are orthogonal if and only if x, yC = .
Proof.
Since vectors x and y are orthogonal in an inner product space, i.
x, y = 0, we have x, yC = ek x, y = ek A 0 = .
Pn the other hand, if x, yC = , then we get x, yC = n k=1 ek x, y = .
This result implies that x, y = 0.
Theorem 3.
Let (Ee2 .
A) be an inner product space and the S-cone inner product on Ee2 -space is defined by x, yC = x, y, then .
x, xC = OuxOuC OuxOuC .
x, yC O OuxOuC OuOuC .
Proof.
we have that .
Since x, yC = k=1 ek x, y, x, k=1 ek x, x k=1 ek OuxOu PnxC = Therefore, k=1 ek OuxOu OuxOu.
OuxOuC OuxOuCP= k=1 ek OuxOu OuxOu, and it means that x, xC = k=1 ek OuxOu OuxOu = OuxOuC OuxOuC .
By triangle inequality of the inner product, we have x, yC = ek x, y O x, x y, y ek (OuxOu OuyOu ) = OuxOuC OuyOuC = (OuxOuC OuyOuC )2 .
Thus, we have that x, yC O OuxOuc OuyOuC .
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Theorem 4.
Let (Ee2 .
A) be an inner product space and the S-cone inner product on Ee2 -space is defined by x, yC = k=1 ek x, y, then .
x, yC w, zC = k=1 ek .
, y w, .
x, yC = x, yC .
If I is an angle between vectors x and y in Ee2 -space, then x, yC = OuxOuC OuyOuC cos I.
And also x, yC = ek x, y = = cos I (Oux, yOuC )2 = ek Oux, y zOuEe2 Oux, yOuEe2 Oux, zOuEe2 ek Oux, yOuEe2 Oux, zOuEe2 = Oux, yOuC Oux, zOuC .
Which means that x y, zC O z, zC y, zC .
Therefore, an Ee2 -space is a cone 2-normed space.
Theorem 5.
Let (Ee2 .
A) be a S-cone inner product on Ee2 space, then ek Oux, yOuEe2 = OuxOuC OuyOuC Oe .
, yC )2 .
Oux, yOuC = x, x y, x x, y y, y ek .
, x y, y Oe .
, .
2 ] ek x, x y, y Oe ek .
, .
2 = x, xC y, yC Oe .
, yC )2 n From the discussion of the cone norm and S-cone inner product, we obtain its properties, among others: additive, multiplication with a scalar and multiplication between two Furthermore, we construct and define a cone 2-norm associated with the S-cone inner product.
Let Ee2 -space be a 2-normed space.
A function OuA.
AOuC :
EeP y Ee2 Ie (Rn .
OuAO.
, be defined by Oux, yOuC = k=1 ek Oux, yOuEe2 , is a cone 2-normed space.
In this case, we call Ee2 -space as cone 2-normed spaces.
The reason for the name can be explained as follows.
For every x, y, z OO Ee2 and OO R, the following statements (CN.
Oux, yOuC = k=1 ek Oux, yOuEe2 O for all x, y OO X, because Oux, yOu PEe2n Ou 0.
(CN.
Oux, yOuC = k=1 ek Oux, yOuEe2 = if and only if Oux, yOuEe2 = 0 as 2-normed space, then Oux, yOuEe2 = 0 if and only if Pxn and y are linearlyPdependent.
(CN.
Oux, yOuC = k=1 ek Oux, yOuEe2 = k=1 ek Ouy, xOuEe2 = Ouy, xOuC .
(CN.
Since Oux, y zOu O Oux, yOu Ouy, zOu, then O Therefore, we have Oux, yOuC = ek Oux, yOuEe2 = OuxOuC OuyOuC Oe .
, yC )2 .
ek OuxOu OuyOu = OuxOuC OuyOuC cos I.
ek (Oux, yOuEe2 )2 = Therefore, we get x, yC = OuxOuC OuyOuC cos I.
Oux, y zOuC = = OuxOuC OuyOuC Oe .
, yC )2 .
x, yC y, yC Proof.
From the definition of the OuA.
AOuC , we have ek OuxOu OuyOu cos I x, xC y, xC Oux, yOuC = Proof.
P x, yC w, zC = k=1 ek x, y k=1 ek w, z = w, .
k=1 ek .
, y .
x, y k=1 ek x, y = k=1 ek x, y = Pn C k=1 ek x, y = x, yC .
Since x, y = OuxOu OuyOu cos I, then In addition, .
, yC )2 = ek x, x y, y Oe ek x, x y, y Oe ek x, y ek x, y y, x = OuxOuC OuyOuC Oe x, yC y, xC .
Since OuxOuC = Oux, xOuC and OuyOuC = y, yC then Oux, yOuC = OuxOuC OuyOuC Oe x, yC y, xC = x, xC y, yC Oe x, yC y, xC .
Here we have Oux, yOuC = x, xC y, xC x, yC y, yC Example 1.
Let (Ee2 .
A) be an inner product space with P a S-cone in R2 .
If we define a function A.
AC : Ee2 y Ee2 Ie (R2 .
OuAO.
by x, yC = .
, y , x, .
, .
then A.
AC is a S-cone inner product in Ee2 -space.
We show that A.
AC satisfies the following properties:
(IC.
Since x, x Ou 0, then x, xC = .
, x , x, .
O for every x OO Ee2 .
Furthermore, x, xC = .
, x , x, .
= if and ony if x, x = 0 if and only if x = 0.
(IC.
x, yC = .
, y , y, .
= y, x, y, x = y, xC for every x, y OO Ee2 .
Thus, we have x, yC = y, xC .
(IC.
From the property of multiplication by scalar, we have x, yC = .
, y , x, .
= .
, y , x, .
= x, yC for every x, y OO Ee2 and OO R.
(IC.
Using triangle inequality of innter product, for every x, y, z OO Ee2 , we have x y, zC = .
y, z , x y, .
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O .
, z y, z , x, z y, .
= .
, z , x, .
, z , y, .
= x, zC y, zC Then, we conclude that A.
A in .
is a S-cone inner product in Ee2 .
Example 2.
Let (Ee2 .
OuA.
AO.
be a standard 2-norm space and (Ee2 .
A)is a S-cone inner product space as in Example 1.
function OuA.
AOuC : Ee2 y Ee2 Ie (R2 .
OuAO.
defined by Oux, yOuC = Oux, yOuEe2 .
Oux, yOuEe2 is a cone 2-norm in Ee2 -space.
We see that it is defined as the standard norms:
x, x x, y = x, x y, y Oe x, y Oux, yOu = y, x y, y = OuxOu OuyOu Oe x, y , and implies that (Oux, yOuC )2 Oux, yOuC Oux, yOuC = (Oux, yOuEe2 .
Oux, yOuEe2 ) A (Oux, yOuEe2 .
Oux, yOuEe2 ) = Oux, yOuEe2 .
Oux, yOuEe2 = .
, x y, y Oe x, y , x, x y, y Oe x, y ) = .
, x y, y Oe x, y y, x , x, x y, y Oe x, y y, .
= .
, x y, y , x, x y, .
Oe .
, y y, x , x, y y, .
= x, xC y, yC Oe x, yC y, xC .
Therefore Oux, yOuC = x, xC y, yC Oe x, yC y, xC .
x, xC x, yC In other word, it means that Oux, yOuC = y, xC y, yC Now, we arrive at the main result of this paper, formulated in the following theorem.
Theorem 6.
Let Oux, yOuC = k=1 ek Oux, yOuEe2 be a cone 2norm and I an angle between vectors x and y in Ee2 -space.
Oux, yOuC = .
Oe cos2 I) OuxOuC OuyOuC .
Proof.
Since Oux, yOuC = k=1 ek Oux, yOuEe2 is a cone 2-norm, x, xC x, yC then we have Oux, yOuC = It means that y, xC y, yC Oux, yOuC = OuxOuC OuyOuC = x, yC y, xC ek OuxOu OuyOu Oe ek x, y y, x ek OuxOu OuOu Oe x, y = .
Oe cos2 I) x, xC y, yC = .
Oe cos2 I) OuxOuC OuyOuC .
Corollary 1.
et Oux, yOuC = k=1 ek Oux, yOuEe2 be a cone 2norm and I an angle between vectors x and y in Ee2 -space.
Then Oux, yOuC = OuxOuC OuyOuC sin I.
In a S-cone inner product space, two vectors x and y are orthogonal if and only if x, yC = .
implies that x, yC = OuxOuC OuyOuC cos I = , and also Oux, yOuC = OuxOuC OuyOuC Oe x, yC , and we have Oux, yOuC = OuxOuC OuyOuC Oe = OuxOuC OuyOuC .
As a conclusion, we have Oux, yOuC = OuxOuC OuyOuC .
Acknowledgement.
This research is supported by the Laboratory of Applied Analysis and Algebra.
Departement of Mathematics.
Institut Teknologi Sepuluh Nopember.
Surabaya.
Indonesia.
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