Copyright © 2012 Fred Greensite. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We present a new proof of the Pythagorean theorem which suggests a particular decomposition of the elements of a topological algebra in terms of an “inverse norm” (addressing unital algebraic structure rather than simply vector space structure). One consequence is the unification of Euclidean norm, Minkowski norm, geometric mean, and determinant, as expressions of this entity in the context of different algebras.