In linear algebra, the transpose 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** or **algebraic adjoint** of a linear map is often used to study the original linear map. This concept is generalised by adjoint functors.

Let denote the algebraic dual space of a vector space
Let and be vector spaces over the same field
If is a linear map, then its **algebraic adjoint** or **dual**,^{[1]} is the map defined by
The resulting functional is called the **pullback** of by

The continuous dual space of a topological vector space (TVS) is denoted by
If and are TVSs then a linear map is **weakly continuous** if and only if in which case we let denote the restriction of to
The map is called the **transpose**^{[2]} or **algebraic adjoint** of
The following identity characterizes the transpose of :^{[3]}
where is the natural pairing defined by

The assignment produces an injective linear map between the space of linear operators from to and the space of linear operators from to If 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 from the category of vector spaces over to itself. One can identify with using the natural injection into the double dual.

- If and are linear maps then
^{[4]} - If is a (surjective) vector space isomorphism then so is the transpose
- If and are normed spaces then

and if the linear operator is bounded then the operator norm of is equal to the norm of ; that is^{[5]}^{[6]}
and moreover,

Suppose now that is a weakly continuous linear operator between topological vector spaces and with continuous dual spaces and respectively.
Let denote the canonical dual system, defined by where and are said to be *orthogonal* if
For any subsets and let
denote the (*absolute*) *polar of in * (resp. *of in *).

- If and are convex, weakly closed sets containing the origin then implies
^{[7]} - If and then
^{[4]}

and

- If and are locally convex then
^{[5]}

Suppose and are topological vector spaces and is a weakly continuous linear operator (so ). Given subsets and define their *annihilators* (with respect to the canonical dual system) by^{[6]}

and

- The kernel of is the subspace of orthogonal to the image of :
^{[7]}

- The linear map is injective if and only if its image is a weakly dense subset of (that is, the image of is dense in when is given the weak topology induced by ).
^{[7]} - The transpose is continuous when both and 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).
^{[8]} - (Surjection of Fréchet spaces): If and are Fréchet spaces then the continuous linear operator is surjective if and only if (1) the transpose is injective, and (2) the image of the transpose of is a weakly closed (i.e. weak-* closed) subset of
^{[9]}

Let be a closed vector subspace of a Hausdorff locally convex space and denote the canonical quotient map by
Assume is endowed with the quotient topology induced by the quotient map
Then the transpose of the quotient map is valued in and
is a TVS-isomorphism onto
If is a Banach space then is also an isometry.^{[6]}
Using this transpose, every continuous linear functional on the quotient space is canonically identified with a continuous linear functional in the annihilator of

Let be a closed vector subspace of a Hausdorff locally convex space
If and if is a continuous linear extension of to then the assignment induces a vector space isomorphism
which is an isometry if is a Banach space.^{[6]}

Denote the inclusion map by The transpose of the inclusion map is whose kernel is the annihilator and which is surjective by the Hahn–Banach theorem. This map induces an isomorphism of vector spaces

If the linear map is represented by the matrix with respect to two bases of and then is represented by the transpose matrix with respect to the dual bases of and hence the name. Alternatively, as is represented by acting to the right on column vectors, 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.

The identity that characterizes the transpose, that is, is formally similar to the 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 *real* 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:^{[10]}

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
^{[4]} - The null space of is the subspace of orthogonal to the range of
^{[4]} - is injective if and only if the range of is weakly closed.
^{[4]}

- Adjoint functors – Relationship between two functors abstracting many common constructions
- Composition operator – Linear operator in mathematics
- Hermitian adjoint – Conjugate transpose of an operator in infinite dimensions
- Riesz representation theorem – Theorem about the dual of a Hilbert space
- Dual space § Transpose of a linear map
- Transpose § Transpose of a linear map

**^**Schaefer & Wolff 1999, p. 128.**^**Trèves 2006, p. 240.**^**Halmos (1974, §44)- ^
^{a}^{b}^{c}^{d}^{e}Schaefer & Wolff 1999, pp. 129–130 - ^
^{a}^{b}Trèves 2006, pp. 240–252. - ^
^{a}^{b}^{c}^{d}Rudin 1991, pp. 92–115. - ^
^{a}^{b}^{c}Schaefer & Wolff 1999, pp. 128–130. **^**Trèves 2006, pp. 199–200.**^**Trèves 2006, pp. 382–383.**^**Trèves 2006, p. 488.

- Halmos, Paul (1974),
*Finite-dimensional Vector Spaces*, Springer, ISBN 0-387-90093-4 - Rudin, Walter (1991).
*Functional Analysis*. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. - Schaefer, Helmut H.; Wolff, Manfred P. (1999).
*Topological Vector Spaces*. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. - Trèves, François (2006) [1967].
*Topological Vector Spaces, Distributions and Kernels*. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.

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