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

Kostadin Trenčevski; Risto Malčeski

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arXiv · 2501.16340

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

Kostadin Trenčevski·Risto Malčeski

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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.

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Kostadin Trenčevski, Risto Malčeski. 2025-01-14. On a generalized n-inner product and the corresponding Cauchy-Schwarz inequality. https://arxiv.org/abs/2501.16340

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