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

A real linear space L of dimension \(\geq n\) is called an n-inner product space if it is equipped with an n-inner product \((a,b/a_ 2,...,a_ n)\), \(a,b,a_ 2,...,a_ n\in L\). 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 \(\| b_ 1,...,b_ n\| \neq 0\). 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


46C50 Generalizations of inner products (semi-inner products, partial inner products, etc.)
46A70 Saks spaces and their duals (strict topologies, mixed topologies, two-norm spaces, co-Saks spaces, etc.)


Zbl 0673.46012
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[1] Diminnie, Demonstratio Math. 6 pp 525– (1973)
[2] Diminnie, Demonstratio Math. 10 pp 169– (1977)
[3] Gähler, Math. Nachr. 40 pp 165– (1969)
[4] Gähler, Math. Nachr. 40 pp 229– (1969)
[5] Gähler, Demonstratio Math. 17 pp 655– (1984)
[6] Misiak, Math. Nachr. 140 pp 299– (1989)
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