definition of the Hermitian adjoint, however, the transpose and the Hermitian adjoint are not the same map. The transpose is a map and is defined for linear maps between any vector spaces and without requiring any additional structure. The Hermitian adjoint maps and is only defined for linear maps between Hilbert spaces, as it is defined in terms of the inner product on the Hilbert space. The Hermitian adjoint therefore requires more mathematical structure than the transpose. However, the transpose is often used in contexts where the vector spaces are both equipped with a nondegenerate bilinear form such as the Euclidean dot product or another inner product. In this case, the nondegenerate bilinear form is often used implicitly to map between the vector spaces and their duals, to express the transposed map as a map For a complex Hilbert space, the inner product is sesquilinear and not bilinear, and these conversions change the transpose into the adjoint map. More precisely: if and are Hilbert spaces and is a linear map then the transpose of and the Hermitian adjoint of which we will denote respectively by and are related. Denote by and the canonical antilinear isometries of the Hilbert spaces and onto their duals. Then is the following composition of maps: Applications to functional analysis Suppose that and are topological vector spaces and that is a linear map, then many of 's properties are reflected in If and are weakly closed, convex sets containing the origin, then implies The null space of is the subspace of orthogonal to the range of is injective if and only if the range of is weakly closed. See also References Bibliography Functional analysis Linear algebra