Fundamental theorem of Hilbert spaces

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.

Antilinear functionals and the anti-dual
Suppose that $H$ is a topological vector space (TVS). A function $f : H → $\mathbb{C}$$ is called semilinear or antilinear if for all $x, y ∈ H$ and all scalars $c$ ,

 Additive: $f (x + y) = f (x) + f (y)$; Conjugate homogeneous: $f (c x) = \overline{c} f (x)$. 

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 $H$).

Pre-Hilbert spaces and sesquilinear forms
A sesquilinear form is a map $B : H × H → $\mathbb{C}$$ such that for all $y ∈ H$, the map defined by $x ↦ B(x, y)$ is linear, and for all $x ∈ H$, the map defined by $y ↦ B(x, y)$ is antilinear. 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 $x ∈ H$; it is called non-negative if $B(x, x) ≥ 0$ for all $x ∈ H$. 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 $x, y ∈ H$.

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. If $B$ is non-negative then it induces a canonical seminorm on $H$, denoted by $$\| \cdot \|$$, defined by $x ↦ B(x, x)^{1/2}$, where if $B$ is also positive definite then this map is a norm. 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 → $\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. 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 $h ∈ H$, we define the canonical maps:


 * $B(h, •) : H → $\mathbb{C}$$ where  $y ↦ B(h, y)$,  and


 * $B(•, h) : H → $\mathbb{C}$$ where  $x ↦ B(x, h)$

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


 * $$H \to \overline{H}^{\prime}$$ defined by  $x ↦ B(x, •)$.

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.

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 $\overline{ f }$ and defined by $x ↦ \overline{ f (x)}$.

Fundamental theorem

 * Fundamental theorem of Hilbert spaces: Suppose that $(H, B)$ is a Hausdorff pre-Hilbert space where $B : H × H → $\mathbb{C}$$ is a sesquilinear form that is linear in its first coordinate and antilinear in its second coordinate. Then the canonical linear mapping from $H$ into the anti-dual space of $H$ is surjective if and only if $(H, B)$ is a Hilbert space, in which case the canonical map is a surjective isometry of $H$ onto its anti-dual.