Analyse Master Cours de Francis Clarke

Mathématiques
Université

Niveau: Supérieur, Master Chapter 7 Hilbert spaces Analyse Master 1 : Cours de Francis Clarke (2011) An inner product on X refers to a bilinear mapping · , · X : X?X ? R (that is, linear separately in each variable) such that x,y X = y,x X ?x, y ? X . A Banach space X is said to be a Hilbert space if it admits an inner product satis- fying x2 = x,x X ?x ? X . Canonical cases of Hilbert spaces include Rn, L2(?), and 2. We have, for exam- ple, u,v Rn = u o v, f ,g L2(?) = ? f (x)g(x)dx. Some rather remarkable consequences follow just from the existence of this scalar product. We suspect that the reader has seen the more immediate ones; we review them now, rather expeditiously. 7.1 Basic properties The first conclusion below is called the Cauchy-Schwarz inequality, and the sec- ond is known as the parallelogram identity. 121

  • then
  • distance between
  • appropriate finite sums
  • then any orthonormal
  • ?y ?m
  • hilbert space
  • conclusion below
  • finite-dimensional subspaces
  • let u?


Clarke, Hilbert space

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