Fundamental theorem of Hilbert spaces

In mathematics, specifically in functional analysis and Hilbert space theory, the fundamental theorem of Hilbert spaces gives a necessary 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 {{mvar|H}} is a topological vector space (TVS).

A function {{math|f : H → $\mathbb{C}$ }} is called semilinear or antilinear{{sfn | Trèves | 2006 | pp=112–123}} if for all {{math|x, y ∈ H}} and all scalars {{mvar|c}} ,

Additive: {{math|1=f (x + y) = f (x) + f (y)}};

Conjugate homogeneous: {{math|1=f (c x) = {{overline|c}} f (x)}}.

The vector space of all continuous antilinear functions on {{mvar|H}} is called the anti-dual space or complex conjugate dual space of {{mvar|H}} and is denoted by $\overline{H}^{\prime}$ (in contrast, the continuous dual space of {{mvar|H}} 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 {{mvar|H}}).{{sfn | Trèves | 2006 | pp=112–123}}

= Pre-Hilbert spaces and sesquilinear forms =

A sesquilinear form is a map {{math|B : H × H → $\mathbb{C}$ }} such that for all {{math|y ∈ H}}, the map defined by {{math|x ↦ B(x, y)}} is linear, and for all {{math|x ∈ H}}, the map defined by {{math|y ↦ B(x, y)}} is antilinear.{{sfn | Trèves | 2006 | pp=112–123}}

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 {{mvar|H}} is called positive definite if {{math|B(x, x) > 0}} for all non-0 {{math|x ∈ H}}; it is called non-negative if {{math|B(x, x) ≥ 0}} for all {{math|x ∈ H}}.{{sfn | Trèves | 2006 | pp=112–123}}

A sesquilinear form {{mvar|B}} on {{mvar|H}} is called a Hermitian form if in addition it has the property that $B(x, y) = \overline{B(y, x)}$ for all {{math|x, y ∈ H}}.{{sfn | Trèves | 2006 | pp=112–123}}

= Pre-Hilbert and Hilbert spaces =

A pre-Hilbert space is a pair consisting of a vector space {{mvar|H}} and a non-negative sesquilinear form {{mvar|B}} on {{mvar|H}};

if in addition this sesquilinear form {{mvar|B}} is positive definite then {{math|(H, B)}} is called a Hausdorff pre-Hilbert space.{{sfn | Trèves | 2006 | pp=112–123}}

If {{mvar|B}} is non-negative then it induces a canonical seminorm on {{mvar|H}}, denoted by $\| \cdot \|$ , defined by {{math|x ↦ B(x, x)^1/2}}, where if {{mvar|B}} is also positive definite then this map is a norm.{{sfn | Trèves | 2006 | pp=112–123}}

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 {{math|B : H × H → $\mathbb{C}$ }} 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 {{mvar|H}}; if {{mvar|H}} is Hausdorff then this completion is a Hilbert space.{{sfn | Trèves | 2006 | pp=112–123}}

A Hausdorff pre-Hilbert space that is complete is called a Hilbert space.

= Canonical map into the anti-dual =

Suppose {{math|(H, B)}} is a pre-Hilbert space. If {{math|h ∈ H}}, we define the canonical maps:

::{{math|B(h, •) : H → $\mathbb{C}$ }} {{space|5}} where {{space|5}} {{math|y ↦ B(h, y)}}, {{space|3}} and

::{{math|B(•, h) : H → $\mathbb{C}$ }} {{space|5}} where {{space|5}} {{math|x ↦ B(x, h)}}

The canonical map{{sfn | Trèves | 2006 | pp=112–123}} from {{mvar|H}} into its anti-dual $\overline{H}^{\prime}$ is the map

:: $H \to \overline{H}^{\prime}$ {{space|5}} defined by {{space|5}} {{math|1=x ↦ B(x, •)}}.

If {{math|(H, B)}} 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 {{math|(H, B)}} is a Hausdorff pre-Hilbert.{{sfn | Trèves | 2006 | pp=112–123}}

There is of course a canonical antilinear surjective isometry $H^{\prime} \to \overline{H}^{\prime}$ that sends a continuous linear functional {{mvar|f}} on {{mvar|H}} to the continuous antilinear functional denoted by {{math|{{overline| f }}}} and defined by {{math|x ↦ {{overline| f (x)}}}}.