The Pythagorean Theorem: I. The finite case.
The Pythagorean Theorem and variants of it are studied. The variations evolve to a formulation in terms of noncommutative, conditional expectations on von Neumann algebras that displays the theorem as the basic result of noncommutative, metric, Euclidean Geometry. The emphasis in the present article is finite dimensionality, both "discrete" and "continuous."
Kadison
Richard V
RV
Mathematics Department, University of Pennsylvania, Philadelphia, PA 19104-6395, USA. kadison@math.upenn.edu