Hilbert spaces and the spectral theorem
K. Chandrasekharan · Texts and readings in mathematics · 1996
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.]