*(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 $(a,b/{a}_{2},\xb7\xb7\xb7,{a}_{n})$, $a,b,{a}_{2},\xb7\xb7\xb7,{a}_{n}\in L$. Every such space has a natural topology defined by the n-norm $\parallel a,{a}_{2},\xb7\xb7\xb7,{a}_{n}\parallel =(a,a|{a}_{2},\xb7\xb7\xb7,{a}_{n})$. 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

where ${b}_{1},\xb7\xb7\xb7,{b}_{n}$ are arbitrary elements of L satisfying $\parallel {b}_{1},\xb7\xb7\xb7,{b}_{n}\parallel \ne 0$. Then theorems on convergence of generalized Fourier series and generalized Parseval equality analogous to those in ordinary inner product spaces can be proved.