On a generalized n-inner product and the corresponding Cauchy-Schwarz inequality

Abstract

In this paper is defined an $n$-inner product of type $⟨{\mathbf{a}}_1,\cdots ,{\mathbf{a}}_n|{\mathbf{b}}_1\cdots {\mathbf{b}}_n⟩$ where ${\mathbf{a}}_1,\cdots ,{\mathbf{a}}_n$, ${\mathbf{b}}_1,\cdots ,{\mathbf{b}}_n$ are vectors from a vector space $V$. This definition generalizes the definition of Misiak of $n$-inner product \cite{2}, such that in special case if we consider only such pairs of sets $\{{\mathbf{a}}_1,\cdots ,{\mathbf{a}}_1\}$ and $\{{\mathbf{b}}_1\cdots {\mathbf{b}}_n\}$ which differ for at most one vector, we obtain the definition of Misiak. The Cauchy-Schwarz inequality for this general type of $n$-inner product is proved and some applications are given.

10 pages