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.

Statement

Preliminaries

Semilinear forms, sesquilinear forms, and the anti-dual

Suppose that H is a topological vector space (TVS). A function L : H → ℂ is called semilinear or antilinear if for all x, y ∈ H and all scalars c, L(x + y) = L(x) + L(y) and L(c x) = ${\overline {c}}$ L(x).^[1] The vector space of all continuous semilinear functions on H is called the anti-dual of H and is denoted by ${\overline {H}}^{\prime }$ (in contrast, the continuous dual space of H is denoted by $H^{\prime }$ ) and we making 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] A sesquilinear form is a map B : H × H → 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 semilinear.^[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 ∈ H; it is called non-negative if B(x, x) ≥ 0 for all x ∈ H.^[1] A sesquilinear form B on H is called a Hermitian form if in addition it has the property that $B(x,y)={\overline {B(x,y)}}$ for all x, y ∈ 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 definition 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 ↦ 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. A Hausdorff pre-Hilbert space that is complete is called a Hilbert space.

Canonical map into the anti-dual

If (H, B) is a pre-Hilbert space then the canonical map from H into its anti-dual ${\overline {H}}^{\prime }$ is the map $H\to {\overline {H}}^{\prime }$ defined by $x\mapsto B(x,\bullet )$ , where $B(x,\bullet ):H\to \mathbb {C}$ is the map defined by y ↦ B(x, y).^[1] 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]