nLab
separable Hilbert space

Contents

Definition

A Hilbert space HH over a field FF of real or complex numbers and with inner product (|)(|) is separable if it has a countable topological base, i. e. a family of vectors e ie_i, iIi\in I where II is at most countable, and such that every vector vHv\in H can be uniquely represented as a series v= iIa ie iv = \sum_{i\in I} a_i e_i where a iFa_i\in F and the sum converges in the norm x=(x|x)\|x\| = \sqrt{(x|x)}.

Revised on April 5, 2013 17:10:46 by Urs Schreiber (131.174.41.18)