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A space of Hilbert is a Espace of Banach (thus complete) of which the standard ∥·∥ rises from a scalar Produit or square ⟨ ·,·⟩ by the formula. It is generalization in unspecified dimension of a Euclidean Espace or square.

Theorem of Mr. Fréchet - J. von Neumann - P. Jordan

A space of Banach (respectively vector Space normalized) is a space of Hilbert (respectively space préhilbertien) if and only if its standard checks the equality

$\backslash |\; X\; +\; there\; \backslash |^2\; +\; \backslash |\; X\; -\; there\; \backslash |^2\; =\; 2\; (\backslash |\; X\; \backslash |^2\; +\; \backslash |\; there\; \backslash |^2)$,

who means that the sum of the squares on sides of parallelogram is equal to the sum of the squares of the diagonals (Règle of the parallelogram).

In the real case the scalar Produit is defined by

$\backslash \; langle\; X,\; there\; \backslash \; rangle\; =\; \backslash \; frac\; \{1\}\; \{4\}\; \backslash \; bigl\; (\backslash |\; X\; +\; there\; \backslash |^2\; -\; \backslash |\; X\; -\; there\; \backslash |^2\; \backslash \; bigr)$.

In the complex case the square Produit is defined by

$\backslash \; langle\; X,\; there\; \backslash \; rangle\; =\; \backslash \; langle\; X,\; there\; \backslash \; rangle\_1+\; I\; \backslash \; langle\; X,\; iy\; \backslash \; rangle\_1$,

where and $I$ is the imaginary Unité (the comlexe number identified with the couple of realities $(0,1)$).

In a space of Hilbert of infinite dimension, the usual concept of bases is replaced by that of Base of Hilbert which allows, either to describe a vector by its coordinates, but to approach it by vector an infinite series having each finished coordinate. One is thus with the confluence of the Linear algebra and the Topologie. It is within the framework of spaces of Hilbert that is developed the theory of the variational Formulation, used in many fields of physics.

In Mechanical quantum, the state of a system is represented by a vector in a space of Hilbert.

Examples of spaces of Hilbert

• ℝ ^N provided with the usual scalar product.

• $L^2\; ()$, space of the functions of summable square with convention which two equal functions almost everywhere equal (see the article on space $L^p\; (\backslash \; Omega)$), are provided with $\backslash \; langle\; F,\; G\; \backslash \; rangle\; =\; \backslash \; int\_a^b\; F\; (X)\; G\; (X)\; \backslash ,\; \backslash \; mathrm\; dx$.
• $l^2$, the space of the continuations of complex numbers such as , le produces scalar of two continuations $u$ and $v$ being by definition the sum of the série$\backslash \; sum\_\; \{n=0\}\; ^\; \backslash \; infty\; u\_n\; \backslash \; overline\; \{v\}\; \_n$

In fact, any space of separable Hilbert is Isomorphe with $l^2$, to see the article on the bases of Hilbert.

See too

• Base of Hilbert
• Theorem of Riesz
• Theorem of projection on convex closed in a space of Hilbert
• Theorem of Lax-Milgram
• Theorem of Stampacchia
• Space of Banach
• functional Analysis
• secondary Measurements

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