Orthogonality and Orthonormality in n‐Inner Product Spaces

Abstract

Introduction This paper is a continuation of investigations an n-inner product spaces given in [S]and an extension of results given in [2] and [5] to arbitrary natural n.Let n be a natural number $: 0, L be a linear space of dimension 2 n and (-, -I ., . . ., .)be a real function on L"+1. In the case n = 1 instead of ( a , . I ., ..., .) we also write (-, a) and (a, b I a,, . . ., a,) is to be understood as the expression (a, b). Let us assume the following conditions: 1. (a,ala,,...,a,) 20, 2. (a, b I 4, (a, a I az, ..., a,) = 0 if and only if a, a*, . .., a, are linearly dependent. * -2 a,) = (b, a I a,, ...) am)-3. (a, b I az, ..., a,) = (a, b I ai,, . . ., ai,) for every permutation (i,, . . ., in) of (2, .. ., n). 4. If n > 1, (a, a I a,, 4, . .., a,) = (a,, a, I a, a3, . . ., %I. 5. (aa, b I a,, ..., a,) = u(a, b I a,, ..., a,) for every real u. 6. (a + a', b I a,, . . ., a,) = (a, b I up, . . ., a,) + (a', b I a,, .. ., am). Then (-, . I ., . . ., .) is called an n-inner product on L and L equipped with the n-inner product is said to be an n-inner product s p e ([S]). The concept of an n-inner product space is a generalization of the concepts of an inner product space (n = I) and of a 2-inner product space ([l], [5]). Every n-inner product apace will be considered to be an n-normed space ([3], [S]) equipped with the n-norm IIa1, a,, * --2 an11 = f(.i, a1 I 4, * --9 am) -Every n-normed space has a natural topology defined by ita n-norm ([4], [S]). This topology on L is locally convex and separated. An n-inner product space d W 8 y S be considered to be a topological space equipped with thie topology. In the m e that L possessea property K the natural topology of L agrees with the topology given by the norm where bl, ..., b. are arbitrary elements of L with Ilbl, ..., b, II =I= 0. *) This article is 8 pert of the dissertation (Technical University Warsaw).

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