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Orthogonality and orthonormality in n-inner product spaces. (English) Zbl 0708.46025

A real linear space L of dimension n is called an n-inner product space if it is equipped with an n-inner product (a,b/a 2 ,···,a n ), a,b,a 2 ,···,a n L. Every such space has a natural topology defined by the n-norm a,a 2 ,···,a n =(a,a|a 2 ,···,a n ). This paper is a continuation of previous investigations of n-inner product spaces by the same author [Math. Nachr. 140, 299-319 (1989; Zbl 0673.46012)]. Here orthogonal and orthonormal sets, generalized Fourier series expansions and representations of n- inner products are studied. Suppose the natural topology of L agrees with the topology given by the norm

a=a,b 2 ,···,b n +b 1 ,a,···,b n +···+b 1 ,b 2 ,···,a,

where b 1 ,···,b n are arbitrary elements of L satisfying b 1 ,···,b n 0. Then theorems on convergence of generalized Fourier series and generalized Parseval equality analogous to those in ordinary inner product spaces can be proved.

Reviewer: I.Vidav

MSC:
46C50Generalizations of inner products
46A70Saks spaces and their duals