In linear algebra and functional analysis, the transpose or algebraic adjoint of a linear map between two vector spaces, defined over the same field, is an induced map between the dual spaces of the two vector spaces. The transpose is often used to study the original linear map. This concept is generalised by adjoint functors.

Definition

denote the algebraic dual space of a vector space be vector spaces over the same field is a linear map, then its algebraic adjoint, or dual, defined by is called the pullback by .

The continuous dual space of a topological vector space is denoted by is weakly continuous if and only if to is called the transpose or algebraic adjoint of . The following identity characterizes the transpose of : is the natural pairing defined by .

Properties

produces an injective to then the space of linear maps is an algebra under composition of maps, and the assignment is then an antihomomorphism of algebras, meaning that . In the language of category theory, taking the dual of vector spaces and the transpose of linear maps is therefore a contravariant functor using the natural injection into the double dual.

• is a (surjective) vector space isomorphism then so is the transpose .

• are normed spaces then

is bounded then the operator norm; that is

• A bounded linear operator between Banach spaces is compact if and only if its adjoint is compact (Schauder's theorem).

Polars

is a weakly continuous linear operator between topological vector spaces and denote the canonical dual system are said to be if and denote the () (resp. ).

• implies .

• then

• are locally convex then

Annihilators

are topological vector spaces is a weakly continuous linear operator (so and , define their (with respect to the canonical dual system) by

and

• The kernel orthogonal to the image of :

  is injective is given the weak topology induced by ).

• are endowed with the weak-* topology (resp. both endowed with the strong dual topology, both endowed with the topology of uniform convergence on compact convex subsets, both endowed with the topology of uniform convergence on compact subsets).

• (Surjection of Fréchet spaces are Fréchet spaces is surjective is injective is a weakly closed (i.e. weak-* closed) subset of .

Duals of quotient spaces

is endowed with the quotient topology induced by the quotient map is a TVS-isomorphism onto is a Banach space is also an isometry. of .

Duals of vector subspaces

be a closed vector subspace of a Hausdorff locally convex space is a Banach space.

Denote the inclusion map and which is surjective by the Hahn–Banach theorem

Representation as a matrix

is represented by the matrix and is represented by the transpose and is represented by the same matrix acting to the left on row vectors. These points of view are related by the canonical inner product on , which identifies the space of column vectors with the dual space of row vectors.

Relation to the Hermitian adjoint

The identity that characterizes the transpose, that is, , is formally similar to the definition of the Hermitian adjoint and 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.

and the Hermitian adjoint of and is the following composition of maps:

Applications to functional analysis

are topological vector spaces's properties are reflected in .

• implies .

• of .

• is weakly closed.

See also

References

Bibliography