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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. 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