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

Abstract

In this paper, we provide an algorithm for verifying the validity of identities of the form $\sum _{A\subseteq \overline{n}}{c}_A{\Vert x_A\Vert }^2=0$, where $x_A=\sum _{i\in A}{x}_i$ 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.

13 pages