A New Orthogonality Relation for Normed Linear Spaces

Abstract

Abetract. A new orthogonality relation for normed linear speaee is introduoedusing a conoept of area of a parallelogram given by E. Silverman. Comparisons are drawn between this relation and an earlier relation used by G. Birkhoff. In addition, this new relation is utilized to obtain new characterizations of inner-product spaces. 1. Introduction. Let (X, 11-11) be a normed linear space of dimension greater than 1. If (X, 11. 1[ ) is an inner-product space with inner-product (.,-), then the most obvious definition of orthogonality is: x 1 y if and only if (x, y) = 0. More general definitiona involving only normed properties were introduced by G.B~SKEOFB in [I] and R. C. JAMES in [?I, [8] and [9]. Research in this area usually concentraha on studyinginteresting properties of such relations and using these properties to find characterizations of innerproduct spaces.The most immediate desirable property of an orthogomlity relation J-is that it should agree with the condition (x, y) = 0 in inner-product spaces. In addition, other properties usually studied are as follows :A. Symmetry: if x 1 y, then y 1 x. B. Homogeneity: if x J-y, then ax 1 by for all real a and b.C. Additivity: if x 1 y and x 1 z, then x 1 (y + z). D. If x, y E X, then there is a real number a such that x 1 (ax + y).The last property is extremely useful in working with an orthogonality relation. Among ot,her things, it guarantees that the relation is not vacuous.To formulate our new orthogonality relation, we will rely on a concept studied by E. SILVESMAN in [13], [14], and [16]. Let F denote the eet of linear functions on X whose norms are less than or equal to 1. Then, defineIntuitively, Ilx, yll may be visualized as the area of the parallelogram with verticee at 0,x, y, and x + y. This quantity was used by F. Smmvm in [16] to examine convexity properties of normed linear spaces. In [6] and [6], S. G&LEB studied this object as an example of a more general notion, which he called a "%norm". He demonstrated that I ] . , has the following properties:1. a) 1 1 4 Yll r 0.b) 1 1 2 , yII = 0 if and only if x and y are dependent. 202 Diminnie, A New Orthogonelity RelationTherefore, ll4 -1 1 2 -YII = llz -Yll which implies that llzll = 1. Sinae this violates the strict convexity of (X, 11. 11 j, no such z and y can exist and The last example illustrates a strictly convex, %dimensional space which satisfies statement 4 of Theorem 2.7 (and hence, statement 6 a h ) but is not an inner-product space.is additive by Lemma 3.4. 0: Example 3.6. Let X = R* with 11. 1[ defined by

Orthogonalities and functional equations

“…For a given semi-inner product and vectors x, y ∈ X we define the semiinner product orthogonality In 1983, Diminnie [56] proposed the following orthogonality relation…”

Banach spaces which embed into their dual

“…In this general context, where there is no inner product, it describes the following geometric property: a vector x is orthogonal to y if each triangle with one side equal to x and another side constructed along y has the third side longer than x. By the way, this is not the unique definition of orthogonality in Banach spaces, but it is surely the oldest and the most intuitive one (see , [Di83], [Ja45] and [Pa86b] for other notions of orthogonality). A simple but important remark is that the classical Riesz representation H ∋ x → f x ∈ H * verifies the property x ∈ Ker(f x )…”

Orthogonal Stability of an Additive-quartic Functional Equation in Non-Archimedean Spaces

“…It is remarkable to note, however, that a real normed space of dimension greater than 2 is an inner product space if and only if the Birkhoff-James orthogonality is symmetric. There are several orthogonality notions on a real normed space such as Birkhoff-James, Boussouis, Singer, Carlsson, unitary-Boussouis, Roberts, Phythagorean, isosceles and Diminnie (see [1]- [3], [7,14,24]). …”