Hilbert spaces and the spectral theorem

Abstract

A Banach space over the complex numbers ℂ, or the real numbers ℝ, is a linear space (over ℂ or ℝ), with a norm ‘‖ ‖’ such that the space is complete with respect to the “metric” d(x, y) = ‖x − y‖ defined by the norm. [A norm is a function ‘‖ ‖”, which is non-negative, and real-valued, with the properties: (i) ‖ax‖ = |a|·‖x‖, a ∈ ℂ; (ii) ‖x + y‖ ≤ ‖x‖ + ‖y‖; (iii) ‖x‖ = 0 ⇔ x = 0.]