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 is a topological vector space (TVS). A function is called semilinear or antilinear if for all and all scalars,
The vector space of all continuous antilinear functions on is called the anti-dual space or complex conjugate dual space of and is denoted by
\overline{H}\prime
H\prime
A sesquilinear form is a map such that for all, the map defined by is linear, and for all, the map defined by 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 is called positive definite if for all non-0 ; it is called non-negative if for all . A sesquilinear form on is called a Hermitian form if in addition it has the property that
B(x,y)=\overline{B(y,x)}
A pre-Hilbert space is a pair consisting of a vector space and a non-negative sesquilinear form on ; if in addition this sesquilinear form is positive definite then is called a Hausdorff pre-Hilbert space. If is non-negative then it induces a canonical seminorm on, denoted by
\| ⋅ \|
Suppose is a pre-Hilbert space. If, we define the canonical maps:
where, and
where
The canonical map from into its anti-dual
\overline{H}\prime
H\to\overline{H}\prime
If 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 is a Hausdorff pre-Hilbert.
There is of course a canonical antilinear surjective isometry
H\prime\to\overline{H}\prime
Fundamental theorem of Hilbert spaces: Suppose that is a Hausdorff pre-Hilbert space where is a sesquilinear form that is linear in its first coordinate and antilinear in its second coordinate. Then the canonical linear mapping from into the anti-dual space of is surjective if and only if is a Hilbert space, in which case the canonical map is a surjective isometry of onto its anti-dual.