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

TL;DR

This paper introduces a generalized n-inner product in vector spaces, extending Misiak's definition, and proves a corresponding Cauchy-Schwarz inequality with some applications.

Contribution

It defines a new generalized n-inner product that broadens previous concepts and establishes a related Cauchy-Schwarz inequality.

Findings

The generalized n-inner product satisfies a Cauchy-Schwarz inequality.

Special cases recover Misiak's n-inner product.

Applications demonstrate the utility of the new definition.

Abstract

In this paper is defined an $n$-inner product of type $\langle {\bf a}_1,\cdots ,{\bf a}_n\vert {\bf b}_1\cdots {\bf b}_n\rangle $ where ${\bf a}_1,\cdots ,{\bf a}_n$, ${\bf b}_1, \cdots ,{\bf b}_n$ 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 $\{ {\bf a}_1,\cdots ,{\bf a}_1\}$ and $\{ {\bf b}_1\cdots {\bf b}_n\}$ 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.