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 → ${\displaystyle \mathbb {C} }$ 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 ${\displaystyle {\overline {H}}^{\prime }}$ (in contrast, the continuous dual space of H is denoted by ${\displaystyle H^{\prime }}$), 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).^[1]

Pre-Hilbert spaces and sesquilinear forms

A sesquilinear form is a map B : H × H → ${\displaystyle \mathbb {C} }$ 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.^[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 ${\displaystyle B(x,y)={\overline {B(y,x)}}}$ 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 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 ${\displaystyle \|\cdot \|}$, 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. The sesquilinear form B : H × H → ${\displaystyle \mathbb {C} }$ 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.^[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 ∈ H, we define the canonical maps:

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

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

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

${\displaystyle H\to {\overline {H}}^{\prime }}$       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.^[1]

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

Fundamental theorem

Fundamental theorem of Hilbert spaces:^[1] Suppose that (H, B) is a Hausdorff pre-Hilbert space where B : H × H → ${\displaystyle \mathbb {C} }$ 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.

See also

• Complex conjugate vector space
• Dual system
• Linear map
• Riesz representation theorem
• Sesquilinear form