Hilbert spaces

Abstract

A Banach space X is called a Hilbert space if it admits a bilinear mapping (x,y) ↦ 〈 x ,y 〉 _X that generates its norm, in the following sense:

$$\|\,x\,\|^{\,2} \: = \:\langle \, x\,, x \,\rangle _X\;~\; \forall \,x\in\, X .$$

The bilinear mapping is then referred to as an inner product on X. Canonical cases of Hilbert spaces include ${\mathbb{R}}^{ n}$, L ²(Ω), and ℓ ². Some rather remarkable consequences for the structure of the space X follow from the mere existence of this scalar product. We begin the chapter with a review of the basic theory of Hilbert spaces. Then we present a smooth minimization principle, in which the differentiability of the Hilbert norm plays a crucial role. The final sections introduce the proximal subdifferential, an important tool in nonsmooth (and nonconvex) analysis.