Suppose that H is a topological vector space (TVS).
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
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.
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.