Hilbert Spaces. Weak Convergence

Abstract

Hilbert spaces are Banach spaces with norm derived from a scalar product. A sequence in a Hilbert space is said to converge weakly if its scalar product with any fixed element of the Hilbert space converges. Weak convergence satisfies important compactness properties that do not hold for ordinary convergence in an infinite dimensional Hilbert space. In particular, any bounded sequence contains a weakly convergent subsequence.