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

$f$ is additive: $f(x+y)=f(x)+f(y)$ ;
$f$ is conjugate homogeneous: $f(c\,x)={\bar {c}}\,f(x)$ .

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 ${\overline {H}}^{\prime }$ . (In contrast, the continuous dual space of $H$ 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 $H$ .)^[1]

Pre-Hilbert spaces and sesquilinear forms

A sesquilinear form is a map $B:H\times H\to \mathbb {C}$ 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 $B$ 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 $B(x,y)={\overline {B(y,x)}}$ 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 $\|\cdot \|$ , 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 $B:H\times H\to \mathbb {C}$ 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 $H$ ; 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,\cdot ):H\mapsto \mathbb {C}

where

y\mapsto B(h,y)

, and

B(\cdot ,h):H\mapsto \mathbb {C}

where

x\mapsto B(x,y)

.

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

H\to {\overline {H}}^{\prime }

defined by

x\mapsto B(x,\cdot )

.

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 $H^{\prime }\to {\overline {H}}^{\prime }$ that sends a continuous linear functional $f$ on $H$ to the continuous antilinear functional denoted by ${\bar {f}}$ and defined by $x\mapsto {\overline {f(x)}}$ .