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.
Suppose that H is a topological vector space (TVS). A function f : H → is called semilinear or antilinear[1] if for all x, y ∈ H and all scalars c ,
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 (in contrast, the continuous dual space of H is denoted by ), 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]
A sesquilinear form is a map B : H × H → 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 for all x, y ∈ H.[1]
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 , 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 → 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.
Suppose (H, B) is a pre-Hilbert space. If h ∈ H, we define the canonical maps:
The canonical map[1] from H into its anti-dual is the map
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 that sends a continuous linear functional f on H to the continuous antilinear functional denoted by f and defined by x ↦ f (x).