International Journal of Mathematics and Mathematical Sciences
Volume 2012 (2012), Article ID 353917, 15 pages
Research Article

A New Proof of the Pythagorean Theorem and Its Application to Element Decompositions in Topological Algebras

Department of Radiological Sciences, University of California Irvine Medical Center, Orange, CA 92868, USA

Received 28 February 2012; Accepted 3 June 2012

Academic Editor: Attila Gilányi

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.


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.