Fundamental theorem of Hilbert spaces

Template: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 Template:Mvar is a topological vector space (TVS). A function Template:Math is called semilinear or antilinearTemplate:Sfn if for all Template:Math and all scalars Template:Mvar ,

• Additive: Template:Math;
• Conjugate homogeneous: Template:Math.

The vector space of all continuous antilinear functions on Template:Mvar is called the anti-dual space or complex conjugate dual space of Template:Mvar and is denoted by $\stackrel{\prime }{\bar{H}}$ (in contrast, the continuous dual space of Template:Mvar is denoted by $H^{\prime }$), 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 Template:Mvar).Template:Sfn

Pre-Hilbert spaces and sesquilinear forms

A sesquilinear form is a map Template:Math such that for all Template:Math, the map defined by Template:Math is linear, and for all Template:Math, the map defined by Template:Math is antilinear.Template: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 Template:Mvar is called positive definite if Template:Math for all non-0 Template:Math; it is called non-negative if Template:Math for all Template:Math.Template:Sfn A sesquilinear form Template:Mvar on Template:Mvar is called a Hermitian form if in addition it has the property that $B(x,y)=\overline{B(y,x)}$ for all Template:Math.Template:Sfn

Pre-Hilbert and Hilbert spaces

A pre-Hilbert space is a pair consisting of a vector space Template:Mvar and a non-negative sesquilinear form Template:Mvar on Template:Mvar; if in addition this sesquilinear form Template:Mvar is positive definite then Template:Math is called a Hausdorff pre-Hilbert space.Template:Sfn If Template:Mvar is non-negative then it induces a canonical seminorm on Template:Mvar, denoted by $\Vert \cdot \Vert$, defined by Template:Math, where if Template:Mvar is also positive definite then this map is a norm.Template: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 Template: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 Template:Mvar; if Template:Mvar is Hausdorff then this completion is a Hilbert space.Template:Sfn A Hausdorff pre-Hilbert space that is complete is called a Hilbert space.

Canonical map into the anti-dual

Suppose Template:Math is a pre-Hilbert space. If Template:Math, we define the canonical maps:

Template:Math Template:Space where Template:Space Template:Math, Template:Space and

Template:Math Template:Space where Template:Space Template:Math

The canonical mapTemplate:Sfn from Template:Mvar into its anti-dual $\stackrel{\prime }{\bar{H}}$ is the map

$H\to \stackrel{\prime }{\bar{H}}$ Template:Space defined by Template:Space Template:Math.

If Template: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 Template:Math is a Hausdorff pre-Hilbert.Template:Sfn

There is of course a canonical antilinear surjective isometry $H^{\prime }\to \stackrel{\prime }{\bar{H}}$ that sends a continuous linear functional Template:Mvar on Template:Mvar to the continuous antilinear functional denoted by Template:Math and defined by Template:Math.

Fundamental theorem

Fundamental theorem of Hilbert spaces:Template:Sfn Suppose that Template:Math is a Hausdorff pre-Hilbert space where Template:Math is a sesquilinear form that is linear in its first coordinate and antilinear in its second coordinate. Then the canonical linear mapping from Template:Mvar into the anti-dual space of Template:Mvar is surjective if and only if Template:Math is a Hilbert space, in which case the canonical map is a surjective isometry of Template: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

• Template:Narici Beckenstein Topological Vector Spaces
• Template:Schaefer Wolff Topological Vector Spaces
• Template:Trèves François Topological vector spaces, distributions and kernels