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

TL;DR

The paper generalizes the classical $n$-inner product by defining a broad $n$-inner product on a real vector space $V$ with $\dim V \ge n$ (Definition 2.1) and establishes its basic axioms, showing it induces an $n$-norm. It proves a Cauchy–Schwarz inequality for this generalized $n$-inner product, $\langle {\bf a}_{1},\dots,{\bf a}_{n}|{\bf b}_{1},\dots,{\bf b}_{n}\rangle^{2} \le \langle {\bf a}_{1},\dots,{\bf a}_{n}|{\bf a}_{1},\dots,{\bf a}_{n}\rangle \langle {\bf b}_{1},\dots,{\bf b}_{n}|{\bf b}_{1},\dots,{\bf b}_{n}\rangle$, with equality characterized by the linear (in)dependence of the vectors or by the subspaces they generate. The framework is then leveraged to define the angle between $n$-dimensional subspaces via $\cos \varphi = \frac{\langle {\bf a}_{1},\dots,{\bf a}_{n}|{\bf b}_{1},\dots,{\bf b}_{n}\rangle}{\|{\bf a}_{1},\dots,{\bf a}_{n}\|\;\|{\bf b}_{1},\dots,{\bf b}_{n}\|}$ and its interpretation as a Grassmann metric, along with induced structures on $\Lambda_{n}(V)$ and a dual $(m-n)$-inner product. Theorem 3.1 further links the angle between subspaces to the angle between their orthogonal complements, with implications for orthogonality and duality and suggesting avenues for further generalization and open questions in related spaces.

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.