An Algorithm for Verifying Some Norm Identities in Inner-Product Spaces

TL;DR

This paper introduces an algorithm to verify complex norm identities in inner-product spaces, enabling the validation of such identities and characterizations of the space's structure.

Contribution

The paper presents a novel algorithm for verifying norm identities in inner-product spaces and demonstrates its use in characterizing these spaces.

Findings

Algorithm successfully verifies various norm identities.

Identities hold exclusively in inner-product spaces.

Provides a method to characterize inner-product spaces.

Abstract

In this paper, we provide an algorithm for verifying the validity of identities of the form $\underset{A\subseteq\overline{n}}{\sum}c_{A}\left\Vert x_{A}\right\Vert ^{2}=0$, where $x_{A}=\underset{i\in A}{\sum}x_{i}$ and $\overline{n}=\{1,...,n\}$ in inner-product spaces. Such algorithm is used to verify the validity, in inner-product spaces, for a number of identities. These include a generalization of the parallelopiped law. We also show that such identities hold only in inner-product spaces. Thus, the algorithm can be used to deduce characterizations of inner-product spaces.