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Erschienen in:
2013 | OriginalPaper | Buchkapitel
7. Hilbert spaces
verfasst von : Francis Clarke
Erschienen in: Functional Analysis, Calculus of Variations and Optimal Control
Verlag: Springer London
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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: 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.
$$\|\,x\,\|^{\,2} \: = \:\langle \, x\,, x \,\rangle _X\;~\; \forall \,x\in\, X .$$