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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 $(a,b/a\sb 2,...,a\sb n)$, $a,b,a\sb 2,...,a\sb n\in L$. Every such space has a natural topology defined by the n-norm $\Vert a,a\sb 2,...,a\sb n\Vert =(a,a\vert a\sb 2,...,a\sb n)$. 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 $$ \Vert a\Vert =\Vert a,b\sb 2,...,b\sb n\Vert +\Vert b\sb 1,a,...,b\sb n\Vert +...+\Vert b\sb 1,b\sb 2,...,a\Vert, $$ where $b\sb 1,...,b\sb n$ are arbitrary elements of L satisfying $\Vert b\sb 1,...,b\sb n\Vert \ne 0$. 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

46C50Generalizations of inner products
46A70Saks spaces and their duals
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