Fundamental theorem of Hilbert spaces

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) = ${\displaystyle {\overline {c}}}$ L(x).[1] The vector space of all continuous semilinear functions on H is called the anti-dual of H and is denoted by ${\displaystyle {\overline {H}}^{\prime }}$ (in contrast, the continuous dual space of H is denoted by ${\displaystyle H^{\prime }}$) 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 ${\displaystyle B(x,y)={\overline {B(x,y)}}}$ 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 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 ${\displaystyle \|\cdot \|}$, 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. A Hausdorff pre-Hilbert space that is complete is called a Hilbert space.

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