Английская Википедия:Fundamental theorem of Hilbert spaces
Шаблон:Short description In mathematics, specifically in functional analysis and Hilbert space theory, the fundamental theorem of Hilbert spaces gives a necessarily and sufficient condition for a Hausdorff pre-Hilbert space to be a Hilbert space in terms of the canonical isometry of a pre-Hilbert space into its anti-dual.
Preliminaries
Antilinear functionals and the anti-dual
Suppose that Шаблон:Mvar is a topological vector space (TVS). A function Шаблон:Math is called semilinear or antilinearШаблон:Sfn if for all Шаблон:Math and all scalars Шаблон:Mvar ,
The vector space of all continuous antilinear functions on Шаблон:Mvar is called the anti-dual space or complex conjugate dual space of Шаблон:Mvar and is denoted by <math>\overline{H}^{\prime}</math> (in contrast, the continuous dual space of Шаблон:Mvar is denoted by <math>H^{\prime}</math>), which we make into a normed space by endowing it with the canonical norm (defined in the same way as the canonical norm on the continuous dual space of Шаблон:Mvar).Шаблон:Sfn
Pre-Hilbert spaces and sesquilinear forms
A sesquilinear form is a map Шаблон:Math such that for all Шаблон:Math, the map defined by Шаблон:Math is linear, and for all Шаблон:Math, the map defined by Шаблон:Math is antilinear.Шаблон:Sfn Note that in Physics, the convention is that a sesquilinear form is linear in its second coordinate and antilinear in its first coordinate.
A sesquilinear form on Шаблон:Mvar is called positive definite if Шаблон:Math for all non-0 Шаблон:Math; it is called non-negative if Шаблон:Math for all Шаблон:Math.Шаблон:Sfn A sesquilinear form Шаблон:Mvar on Шаблон:Mvar is called a Hermitian form if in addition it has the property that <math>B(x, y) = \overline{B(y, x)}</math> for all Шаблон:Math.Шаблон:Sfn
Pre-Hilbert and Hilbert spaces
A pre-Hilbert space is a pair consisting of a vector space Шаблон:Mvar and a non-negative sesquilinear form Шаблон:Mvar on Шаблон:Mvar; if in addition this sesquilinear form Шаблон:Mvar is positive definite then Шаблон:Math is called a Hausdorff pre-Hilbert space.Шаблон:Sfn If Шаблон:Mvar is non-negative then it induces a canonical seminorm on Шаблон:Mvar, denoted by <math>\| \cdot \|</math>, defined by Шаблон:Math, where if Шаблон:Mvar is also positive definite then this map is a norm.Шаблон:Sfn This canonical semi-norm makes every pre-Hilbert space into a seminormed space and every Hausdorff pre-Hilbert space into a normed space. The sesquilinear form Шаблон:Math is separately uniformly continuous in each of its two arguments and hence can be extended to a separately continuous sesquilinear form on the completion of Шаблон:Mvar; if Шаблон:Mvar is Hausdorff then this completion is a Hilbert space.Шаблон:Sfn A Hausdorff pre-Hilbert space that is complete is called a Hilbert space.
Canonical map into the anti-dual
Suppose Шаблон:Math is a pre-Hilbert space. If Шаблон:Math, we define the canonical maps:
- Шаблон:Math Шаблон:Space where Шаблон:Space Шаблон:Math, Шаблон:Space and
The canonical mapШаблон:Sfn from Шаблон:Mvar into its anti-dual <math>\overline{H}^{\prime}</math> is the map
- <math>H \to \overline{H}^{\prime}</math> Шаблон:Space defined by Шаблон:Space Шаблон:Math.
If Шаблон:Math is a pre-Hilbert space then this canonical map is linear and continuous; this map is an isometry onto a vector subspace of the anti-dual if and only if Шаблон:Math is a Hausdorff pre-Hilbert.Шаблон:Sfn
There is of course a canonical antilinear surjective isometry <math>H^{\prime} \to \overline{H}^{\prime}</math> that sends a continuous linear functional Шаблон:Mvar on Шаблон:Mvar to the continuous antilinear functional denoted by Шаблон:Math and defined by Шаблон:Math.
Fundamental theorem
- Fundamental theorem of Hilbert spaces:Шаблон:Sfn Suppose that Шаблон:Math is a Hausdorff pre-Hilbert space where Шаблон:Math is a sesquilinear form that is linear in its first coordinate and antilinear in its second coordinate. Then the canonical linear mapping from Шаблон:Mvar into the anti-dual space of Шаблон:Mvar is surjective if and only if Шаблон:Math is a Hilbert space, in which case the canonical map is a surjective isometry of Шаблон:Mvar onto its anti-dual.
See also
- Complex conjugate vector space
- Dual system
- Hilbert space
- Pre-Hilbert space
- Linear map
- Riesz representation theorem
- Sesquilinear form
References
- Шаблон:Narici Beckenstein Topological Vector Spaces
- Шаблон:Schaefer Wolff Topological Vector Spaces
- Шаблон:Trèves François Topological vector spaces, distributions and kernels
