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 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.

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Preliminaries

Antilinear functionals and the anti-dual

Suppose that H is a topological vector space (TVS). A function f : H → C {\displaystyle \mathbb {C} } is called semilinear or antilinear ¹ if for all x, y ∈ 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 H ¯ ′ {\displaystyle {\overline {H}}^{\prime }} (in contrast, the continuous dual space of H is denoted by H ′ {\displaystyle 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).²

Pre-Hilbert spaces and sesquilinear forms

A sesquilinear form is a map B : H × H → C {\displaystyle \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.³ 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.⁴ A sesquilinear form B on H is called a Hermitian form if in addition it has the property that B ( x , y ) = B ( y , x ) ¯ {\displaystyle B(x,y)={\overline {B(y,x)}}} for all x, y ∈ H.⁵

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.⁶ If B is non-negative then it induces a canonical seminorm on H, denoted by ‖ ⋅ ‖ {\displaystyle \|\cdot \|} , defined by x ↦ B(x, x)1/2, where if B is also positive definite then this map is a norm.⁷ 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 → C {\displaystyle \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.⁸ 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 → C {\displaystyle \mathbb {C} } where y ↦ B(h, y), and B(•, h) : H → C {\displaystyle \mathbb {C} } where x ↦ B(x, h)

The canonical map⁹ from H into its anti-dual H ¯ ′ {\displaystyle {\overline {H}}^{\prime }} is the map

H → H ¯ ′ {\displaystyle 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.¹⁰

There is of course a canonical antilinear surjective isometry H ′ → H ¯ ′ {\displaystyle H^{\prime }\to {\overline {H}}^{\prime }} that sends a continuous linear functional f on H to the continuous antilinear functional denoted by f and defined by x ↦ f (x).