Fundamental theorem of Hilbert spaces

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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 $H$ is a topological vector space (TVS). A function $f : H → [math]\displaystyle{ \mathbb{C} }[/math]$ is called semilinear or antilinear^[1] if for all $x, y \in H$ and all scalars $c$ ,

Additive: $f (x + y) = f (x) + f (y)$ ;
Conjugate homogeneous: $f (c x) = c f (x)$ .

The vector space of all continuous antilinear functions on $H$ is called the anti-dual space or complex conjugate dual space of $H$ and is denoted by [math]\displaystyle{ \overline{H}^{\prime} }[/math] (in contrast, the continuous dual space of $H$ is denoted by [math]\displaystyle{ 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 $H$ ).^[1]

Pre-Hilbert spaces and sesquilinear forms

A sesquilinear form is a map $B : H × H → [math]\displaystyle{ \mathbb{C} }[/math]$ such that for all $y \in H$ , the map defined by $x \mapsto B (x, y)$ is linear, and for all $x \in H$ , the map defined by $y \mapsto B (x, y)$ is antilinear.^[1] 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 $H$ is called positive definite if $B (x, x) > 0$ for all non-0 $x \in H$ ; it is called non-negative if $B (x, x) \geq 0$ for all $x \in H$ .^[1] A sesquilinear form $B$ on $H$ is called a Hermitian form if in addition it has the property that [math]\displaystyle{ B(x, y) = \overline{B(y, x)} }[/math] for all $x, y \in H$ .^[1]

Pre-Hilbert and Hilbert spaces

A pre-Hilbert space is a pair consisting of a vector space $H$ and a non-negative sesquilinear form $B$ on $H$ ; if in addition this sesquilinear form $B$ is positive definite then $(H, B)$ is called a Hausdorff pre-Hilbert space.^[1] If $B$ is non-negative then it induces a canonical seminorm on $H$ , denoted by [math]\displaystyle{ \| \cdot \| }[/math], defined by $x \mapsto B (x, x) 1/2$ , where if $B$ is also positive definite then this map is a norm.^[1] 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 $B : H × H → [math]\displaystyle{ \mathbb{C} }[/math]$ is separately uniformly continuous in each of its two arguments and hence can be extended to a separately continuous sesquilinear form on the completion of $H$ ; if $H$ is Hausdorff then this completion is a Hilbert space.^[1] A Hausdorff pre-Hilbert space that is complete is called a Hilbert space.

Canonical map into the anti-dual

Suppose $(H, B)$ is a pre-Hilbert space. If $h \in H$ , we define the canonical maps:

B(h, •) : H → [math]\displaystyle{ \mathbb{C} }[/math]

where

y \mapsto B (h, y)

, and

B(•, h) : H → [math]\displaystyle{ \mathbb{C} }[/math]

where

x \mapsto B (x, h)

The canonical map^[1] from $H$ into its anti-dual [math]\displaystyle{ \overline{H}^{\prime} }[/math] is the map

[math]\displaystyle{ H \to \overline{H}^{\prime} }[/math] defined by

x \mapsto B (x, •)

.

If $(H, B)$ 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 $(H, B)$ is a Hausdorff pre-Hilbert.^[1]

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