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

A real linear space L of dimension $\ge n$ is called an n-inner product space if it is equipped with an n-inner product $\left(a,b/{a}_{2},···,{a}_{n}\right)$, $a,b,{a}_{2},···,{a}_{n}\in L$. Every such space has a natural topology defined by the n-norm $\parallel a,{a}_{2},···,{a}_{n}\parallel =\left(a,a|{a}_{2},···,{a}_{n}\right)$. 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

$\parallel a\parallel =\parallel a,{b}_{2},···,{b}_{n}\parallel +\parallel {b}_{1},a,···,{b}_{n}\parallel +···+\parallel {b}_{1},{b}_{2},···,a\parallel ,$

where ${b}_{1},···,{b}_{n}$ are arbitrary elements of L satisfying $\parallel {b}_{1},···,{b}_{n}\parallel \ne 0$. 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:
 46C50 Generalizations of inner products 46A70 Saks spaces and their duals