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In mathematics, specifically in [[Functional analysis|functional analysis]] and [[Hilbert space]] theory, the '''fundamental theorem of Hilbert spaces''' gives a necessarily and sufficient condition for a [[Hausdorff space|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|topological vector space]] (TVS). 
A function {{math|''f'' : ''H'' → <math>\mathbb{C}</math>}} is called '''[[Semilinear map|semilinear]]''' or '''[[Antilinear map|antilinear]]'''{{sfn | Trèves | 2006 | pp=112–123}}  if for all {{math|''x'', ''y'' ∈ ''H''}} and all scalars {{mvar|c}} ,

<ul>
<li>'''[[Additive map|Additive]]''': {{math|1=''f'' (''x'' + ''y'') = ''f'' (''x'') + ''f'' (''y'')}};</li>
<li>'''[[Antilinear map|Conjugate homogeneous]]''': {{math|1=''f'' (''c'' ''x'') = {{overline|''c''}} ''f'' (''x'')}}.</li>
</ul>

The vector space of all continuous antilinear functions on {{mvar|H}} is called the anti-dual space or [[Complex conjugate vector space|complex conjugate dual space]] of {{mvar|H}} and is denoted by <math>\overline{H}^{\prime}</math> (in contrast, the continuous dual space of {{mvar|H}} 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 [[Operator norm|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|sesquilinear form]] is a map {{math|''B'' : ''H'' × ''H'' → <math>\mathbb{C}</math>}} such that for all {{math|''y'' ∈ ''H''}}, the map defined by {{math|''x'' ↦ ''B''(''x'', ''y'')}} is [[Linear map|linear]], and for all {{math|''x'' ∈ ''H''}}, the map defined by {{math|''y'' ↦ ''B''(''x'', ''y'')}} is [[Semilinear map|antilinear]].{{sfn | Trèves | 2006 | pp=112–123}}  
Note that in [[Physics:Physics|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 <math>B(x, y) = \overline{B(y, x)}</math> 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|seminorm]] on {{mvar|H}}, denoted by <math>\| \cdot \|</math>, defined by {{math|''x'' ↦ ''B''(''x'', ''x'')<sup>1/2</sup>}}, where if {{mvar|B}} is also positive definite then this map is a [[Norm (mathematics)|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'' → <math>\mathbb{C}</math>}} is separately uniformly continuous in each of its two arguments and hence can be extended to a separately continuous sesquilinear form on the [[Complete topological vector space|completion]] of {{mvar|H}}; if {{mvar|H}} is [[Hausdorff space|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'' → <math>\mathbb{C}</math>}} {{space|5}} where {{space|5}} {{math|''y'' ↦ ''B''(''h'', ''y'')}}, {{space|3}} and

::{{math|''B''(•, ''h'') : ''H'' → <math>\mathbb{C}</math>}} {{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 <math>\overline{H}^{\prime}</math> is the map 

::<math>H \to \overline{H}^{\prime}</math> {{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|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 <math>H^{\prime} \to \overline{H}^{\prime}</math> 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'')}}}}.

== Fundamental theorem ==

:'''Fundamental theorem of Hilbert spaces''':{{sfn | Trèves | 2006 | pp=112–123}}  Suppose that {{math|(''H'', ''B'')}} is a [[Hausdorff space|Hausdorff]] pre-Hilbert space where {{math|''B'' : ''H'' × ''H'' → <math>\mathbb{C}</math>}} is a [[Sesquilinear form|sesquilinear form]] that is linear in its first coordinate and antilinear in its second coordinate. Then the canonical linear mapping from {{mvar|H}} into the anti-dual space of {{mvar|H}} is surjective if and only if {{math|(''H'', ''B'')}} is a Hilbert space, in which case the canonical map is a surjective [[Isometry|isometry]] of {{mvar|H}} 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 ==
{{reflist}}

* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn | Narici | 2011 | p=}} -->
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn | Schaefer | 1999 | p=}} -->
* {{Trèves François Topological vector spaces, distributions and kernels}} <!-- {{sfn | Trèves | 2006 | p=}} -->

{{Functional Analysis}}

[[Category:Topological vector spaces]]

{{Sourceattribution|Fundamental theorem of Hilbert spaces}}