Download Characterization of 2-inner Product by Strictly Convex 2

International Journal of Mathematical Analysis
Vol. 8, 2014, no. 33, 1647 - 1652
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ijma.2014.46174
Characterization of 2-inner Product by Strictly
Convex 2-Norm of Module c
Risto Malčeski
Faculty of informatics, FON University
Bul. Vojvodina bb, 1000 Skopje, Macedonia
Katerina Anevska
Faculty of informatics, FON University
Bul. Vojvodina bb, 1000 Skopje, Macedonia
Copyright © 2014 Risto Malčeski and Katerina Anevska. This is an open access article
distributed under the Creative Commons Attribution License, which permits unrestricted use,
distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Exploring and finding the necessary and sufficient conditions 2-normed
space be a 2-pre-Hilbert space, as a problem is focus of researching of many
mathematicians. Some of the characterizations of 2-inner product are noted in
[2], [6], [9] and [11]. In this paper we will give the term of strictly convex
norm with positive module c, and will use that norm to characterize an 2-inner
product.
Mathematics Subject Classification. 46C50, 46C15, 46B20
Keywords: 2-inner product, 2-norm, strictly convex norm of module c, strictly
convex normed space
1
Introduction
Let L be a real vector space with dimension greater than 1 and || ⋅, ⋅ || be a real
function on L × L which satisfies the following:
1648
Risto Malčeski and Katerina Anevska
|| x, y ||≥ 0 , for every x, y ∈ L и || x, y ||= 0 if and only if the set {x, y}
is linearly dependent;
b) || x, y ||=|| y , x ||, for every x, y ∈ L ;
c) || α x, y ||=| α | ⋅ || x, y ||, for every x, y ∈ L and for every α ∈ R ;
d) || x + y , z ||≤|| x, z || + || y , z ||, for every x, y , z ∈ L.
The function || ⋅, ⋅ || is called as 2-norm on L, and ( L,|| ⋅, ⋅ ||) is called as vector
2-normed space. ([13]).
a)
Let n > 1 be a natural number, L be a real vector space, dim L ≥ n and
(⋅, ⋅ | ⋅) be a real function on L × L × L such that:
i) ( x, x | y ) ≥ 0 , for every x, y ∈ L и ( x, x | y ) = 0 if and only if a and
b are linearly dependent ;
ii) ( x, y | z ) = ( y , x | z ) , for every x, y , z ∈ L ,
iii) ( x, x | y ) = ( y , y | x ) , for every x, y ∈ L ;
iv) (α x, y | z ) = α ( x, y | z ) , for every x, y , z ∈ L. and for every α ∈ R ; and
v) ( x + x1, y | z ) = ( x, y | z ) + ( x1, y | z ) , for every x1, x, y , z ∈ L .
The function (⋅, ⋅ | ⋅) is called as 2-inner product, and ( L,(⋅, ⋅ | ⋅)) is called as
2-pre-Hilbert space ([2]).
Concepts of 2-norm and 2-inner product are two-dimensional analogies of the
concepts of a norm and an inner product. R. Ehret proved ([11]), if ( L,(⋅, ⋅ | ⋅)) be
a 2-pre-Hilbert space, then the equality
|| x, y ||= ( x, x | y )1/2 ,
(1)
for every x, y ∈ L dеfines a 2-norm, and so, we get vector 2-normed space
( L,|| ⋅, ⋅ ||) in which for every x, y , z ∈ L hold the following equality :
|| x + y , z||2 −||x − y , z||2
4
2
2
( x, y | z ) =
,
(2)
|| x + y , z || + || x − y , z || = 2(|| x, z ||2 + || y , z ||2 ) ,
(3)
In fact, equality (3) is two-dimensional analogy of the parallelogram equality, and
is called as parallelepiped equality ([9]). Further, if ( L,|| ⋅, ⋅ ||) is vector 2-normed
space, such that for every x, y , z ∈ L holds (3), than (2) defines 2-inner product
on L , and again (1) holds.
In [2] C. Diminnie and A. White gave characterization of 2-pre-Hilbert space
by partial derivatives of 2-functionals, i.e. prove the following: if ( L,(⋅, ⋅ | ⋅)) be a
2-pre-Hilbert space in which the norm is defined by (1), then for every x, y , z ∈ L
holds the following equality
||x + ty , z ||−|| x , z||
.
2t
t →0
( x, y | z ) = lim
1649
Characterization of 2-inner product
2
Strictly convex and middle strictly convex
2-norm of module c
Definition 1. Let ( L,|| ⋅, ⋅ ||) be a real 2-normed space and c > 0 . The 2-norm
|| ⋅, ⋅ || is called as strictly convex of module c if
|| tx + (1 − t ) y, z ||2 ≤ t || x, z ||2 + (1 − t ) || y , z ||2 − ct (1 − t ) || x − y, z ||2 ,
(4)
for every x, y , z ∈ L , z ∉V ( x, y ) and for every t ∈ (0,1) . The 2-norm || ⋅, ⋅ || is
called as middle strictly convex of module c if the inequality (4) is satisfied only
for t = 12 , i.e. if
x+ y
||x , z ||2 +|| y , z||2
2
|| 2 , z ||2 ≤
for every x, y , z ∈ L , z ∉V ( x, y ) .
− 4c || x − y , z ||2 ,
(5)
Remark 1. By definition 1, is clear that each strictly convex norm of module
c is middle strictly convex of the same module c , and if 2-norm is not middle
strictly convex of module c , then that norm is not strictly convex of the same
module c .
Example 1. On the set of the bounded real sequences l ∞ ,
xi x j
|| x, y ||= sup
, x = ( xi )i∞=1, y = ( yi )i∞=1 ∈ l ∞
i , j∈N yi y j
(6)
i< j
∞
defines 2-norm, and (l ,|| ⋅, ⋅ ||) is a real 2-normed space ([1]). The vectors
x = (1 − 12 ,1 − 12 ,...,1 − 1n ,...) , y = (0,1 − 12 ,1 − 12 ,...,1 − n1−1 ,...) and
2
z = (1,0,0,...,0,...)
2
2
2
hold
x+ y
|| x, z ||=|| y , z ||=|| 2 , z ||= 1 , || x − y , z ||= 12
2
and z ∉V ( x, y ) . Thus, for every c > 0 is true that
|| x , z ||2 +|| y , z||2
x+ y
c =
|| 2 , z ||2 = 1 > 1 − 64
− 4c || x − y , z ||2 .
2
It means, the 2-norm (6) is not middle strictly convex of module c , and thus, by
Remark 1 that norm is not strictly convex of module c . ■
Example 2. Let ( L,(⋅, ⋅)) be a real pre-Hilbert space. Then,
|| x, y ||=
( x, x ) ( x, y )
1/2
,
(7)
( y, x) ( y, y )
for every x, y ∈ L defines a standard 2-norm, i.e. ( L,|| ⋅, ⋅ ||) is a real 2-normed
space. It’s easy to check validity of the following for c = 1
1650
Risto Malčeski and Katerina Anevska
|| tx + (1 − t ) y , z ||2 ≤ t || x, z ||2 +(1 − t ) || y , z ||2 −t (1 − t ) || x − y , z ||2 .
It means, the 2-norm (7) is strictly convex of module 1, and thus, that norm is
middle strictly convex of module 1. ■
Let x, y ∈ L are non-zero elements and V ( x, y ) is subspace of L
generated by the vectors x and y . The vector 2-normed space ( L,|| ⋅, ⋅ ||) is
x+ y
called as strictly convex if || x, z ||=|| y , z ||=|| 2 , z ||= 1 and z ∉V ( x, y ) , for
x, y , z ∈ L , imply x = y ([4]). The other characterizations of strictly convex
2-normed spaces, are given in [3], [4], [7], [8], [10] and [15].
Theorem 1. If ( L,|| ⋅, ⋅ ||) be a real 2-normed space with middle strictly
convex norm of module c > 0 , then L is strictly convex.
Proof. Let L is a real 2-normed space with 2-norm middle strictly convex of
x+ y
module c > 0 . If || x, z ||=|| y , z ||=|| 2 , z ||= 1 , for x, y , z ∈ L and z ∉V ( x, y ) ,
then by (5) follows || x − y , z ||2 ≤ 0 , i.e. || x − y , z ||= 0 . But, by the Axiom а) of
Definition of 2-norm, the last equality is satisfied if and only if the set {x − y , z}
is linearly dependent, and by z ∉V ( x, y ) , we get x = y , i.e. L is strictly
convex. ■
Remark 2. By Remark 1 and Theorem 1, follows each real 2-normed space
with strictly convex norm by module c > 0 is strictly convex.
3
Characterization of 2-inner product
By Example 2 we get, 2-norm (7) is strictly convex of module 1. On the other
hand, in [4] is proved that
( x, y ) ( x, z )
( x, y | z ) =
( z, y ) ( z, z )
defines 2-inner product, in which the norm (7) is defined by (1). The last, is direct
reason for giving the following characterization of 2-inner product by strictly
convex and middle strictly convex norm of module 1.
Theorem 2. Let ( L,|| ⋅, ⋅ ||) be a real 2-normed space. The following states are
equivalent:
1) 2-norm || ⋅, ⋅ || is strictly convex of module 1;
2) 2-norm || ⋅, ⋅ || is middle strictly convex of module 1;
3) L is 2-pre-Hilbert space.
Proof. We’ll prove the following sequence of implications: 1) ⇒ 2) ⇒ 3)
⇒ 1) .
Characterization of 2-inner product
1651
1) ⇒ 2). Holds, by Remark 1.
2) ⇒ 3). Let the 2-norm is middle strictly convex of module c = 1 . Therefore
(5) implies the following inequality
x+ y
|| 2 , z ||2 ≤
||x , z ||2 +|| y , z||2 1
− 4 || x − y , z ||2
2
and thus,
|| x + y , z ||2 + || x − y , z ||2 ≤ 2(|| x, z ||2 + || y, z ||2 ) .
for every x, y , z ∈ L . Letting u = x + y and v = x − y in (8) we get
(8)
2(|| u, z ||2 + || v, z ||2 ) ≤|| u + v, z ||2 + || u − v, z ||2 ,
(9)
for every u, v, z ∈ L . Finally, the inequalities (8) and (9) imply the equality (3), i.e.
(2) defines 2-inner product on L , and furthermore holds (1).
3) ⇒ 1) . Let exist 2-inner product (⋅, ⋅ | ⋅) in which the 2-norm is defined by
(1). Then,
|| tx + (1 − t ) y , z ||2 = (tx + (1 − t ) y , tx + (1 − t ) y | z )
= t 2 ( x, x | z ) + 2t (1 − t )( x, y | z ) + (1 − t )2 ( y , y | z )
= t ( x, x | z ) − t (1 − t )( x, x | z ) + 2t (1 − t )( x, y | z ) + (1 − t )( y , y | z ) − t (1 − t )( y , y | z )
= t ( x, x | z ) + (1 − t )( y , y | z ) − t (1 − t )[( x, x | z ) − 2( x, y | z ) + ( y , y | z )]
= t ( x, x | z ) + (1 − t )( y , y | z ) − t (1 − t )( x − y , x − y | z )
≤ t || x, z ||2 +(1 − t ) || y, z ||2 −t (1 − t ) || x − y , z ||2 .
It means that the 2-norm || ⋅, ⋅ || is strictly convex of module 1. ■
Remark 3. Example 1 shows the following state: for every c > 0 the space
(l ∞ ,|| ⋅, ⋅ ||) in which 2-norm is given by (6) is not strictly convex of module c .
Now, by Theorem 2, there isn’t any 2-inner product which provides the 2-norm
(6).
References
[1] A. Malčeski, l∞ as n-normed space, Matematički bilten, Vol. 21 (XXI) (1997),
103-110.
[2] C. Diminnie and A. White, 2-Inner Product Spaces and Gâteaux partial
derivatives, Comment Math. Univ. Carolinae, 16(1) (1975), 115-119.
[3] C. Diminnie and A. White, Strict convexity in topological vector spaces,
Math. Japon., 22 (1977), 49-56.
[4] C. Diminnie and S. Gahler, Some geometric results concerning strictly
2-convex 2-normed spaces, Math. Seminar Notes, Kobe Univ., 6 (1978),
245-253.
1652
Risto Malčeski and Katerina Anevska
[5] C. Diminnie, S. Gähler and A. White, 2-Inner Product Spaces II,
Demonstratio Mathematica, Vol. X, No 1 (1977), 169 – 188.
[6] C. Diminnie, S. Gähler and A. White, 2-Inner Product Spaces, Demonstratio
Mathematica, Vol. VI (1973), 525 – 536.
[7] C. Diminnie, S. Gahler and A. White, Remarks on Strictly Convex and
Strictly 2-Convex 2-Normed Spaces, Math.Nachr., 88 (1979), 363-372.
[8] C. Diminnie, S. Gahler and A. White, Strictly Convex Linear 2-Normed
Spaces, Math. Nachr., 59 (1974), 319-324.
[9] K. Anevska and R. Malčeski, Characterization of 2-inner product using
Euler-Lagrange type of equality, International Journal of Science and
Research (IJSR), ISSN 2319-7064, Vol. 3 Iss. 6 (2014), 1220-1222.
[10] K.S. Ha, Y.J Cho and A. White, Strictly convex and strictly 2-convex linear
2- normed spaces, Math.Japon., 33 (1988), 375-384.
[11] R. Ehret, Linear 2-normed Spaces, Doctoral Diss., Saint Louis Univ., 1969.
[12] R.W. Freese, Y.J. Cho and S.S. Kim, Stryctly 2-convex linear 2-normed
spaces, J. Korean Math. Soc., 29 (1992), 391-400.
[13] S. Gähler, Lineare 2-normierte Räume, Math. Nachr. 28 (1965), 1-42.
[14] Y.J. Cho and S.S. Kim, Gâteaux derivatives and 2-Inner Product Spaces,
Glasnik matematički, Vol. 27(47) (1992), 271 – 282.
[15] Y.J. Cho, K.S. Ha and S.W. Kim, Strictly Convex Linear 2-Normed Spaces,
Math. Japonica, 26 (1981), 475-478.
Received: June 19, 2014