Inlinear algebra, the transpose of alinear map between two vector spaces, defined over the samefield, is an induced map between thedual spaces of the two vector spaces. Thetranspose oralgebraic adjoint of a linear map is often used to study the original linear map. This concept is generalised byadjoint functors.

Let${\displaystyle X^{\mathrm{\#}}}$ denote thealgebraic dual space of a vector space⁠${\displaystyle X}$⁠. Let${\displaystyle X}$ and${\displaystyle Y}$ be vector spaces over the same field⁠${\displaystyle \mathcal{K}}$⁠. If${\displaystyle u:X\to Y}$ is alinear map, then itsalgebraic adjoint ordual,^[1] is the map${\displaystyle {}^{\mathrm{\#}}u:Y^{\mathrm{\#}}\to X^{\mathrm{\#}}}$ defined by⁠${\displaystyle f\mapsto f\circ u}$⁠. The resulting functional${\displaystyle {}^{\mathrm{\#}}u(f):=f\circ u}$ is called thepullback of${\displaystyle f}$ by⁠${\displaystyle u}$⁠.

Thecontinuous dual space of atopological vector space (TVS)${\displaystyle X}$ is denoted by⁠${\displaystyle X^{\mathrm{\prime }}}$⁠. If${\displaystyle X}$ and${\displaystyle Y}$ are TVSs then a linear map${\displaystyle u:X\to Y}$ isweakly continuous if and only if⁠${\displaystyle {}^{\mathrm{\#}}u\left(Y^{\mathrm{\prime }}\right)\subseteq X^{\mathrm{\prime }}}$⁠, in which case we let${\displaystyle {}^{\text{t}}u:Y^{\mathrm{\prime }}\to X^{\mathrm{\prime }}}$ denote the restriction of${\displaystyle {}^{\mathrm{\#}}u}$ to⁠${\displaystyle Y^{\mathrm{\prime }}}$⁠. The map${\displaystyle {}^{\text{t}}u}$ is called thetranspose^[2] oralgebraic adjoint of⁠${\displaystyle u}$⁠. The following identity characterizes the transpose of⁠${\displaystyle u}$⁠:^[3]$${\displaystyle ⟨{}^{\text{t}}u(f),x⟩=⟨f,u(x)⟩\text{ for all }f\in Y^{\mathrm{\prime }}\text{ and }x\in X,}$$where${\displaystyle ⟨\cdot ,\cdot ⟩}$ is thenatural pairing defined by⁠${\displaystyle 1}$⁠.

The assignment${\displaystyle u\mapsto {}^{\text{t}}u}$ produces aninjective linear map between the space of linear operators from${\displaystyle X}$ to${\displaystyle Y}$ and the space of linear operators from${\displaystyle Y^{\mathrm{\#}}}$ to⁠${\displaystyle X^{\mathrm{\#}}}$⁠. If${\displaystyle X=Y}$ then the space of linear maps is analgebra undercomposition of maps, and the assignment is then anantihomomorphism of algebras, meaning that⁠${\displaystyle 1}$⁠. In the language ofcategory theory, taking the dual of vector spaces and the transpose of linear maps is therefore acontravariant functor from the category of vector spaces over${\displaystyle \mathcal{K}}$ to itself. One can identify${\displaystyle {}^{\text{t}}\left({}^{\text{t}}u\right)}$ with${\displaystyle u}$ using the natural injection into the double dual.

• If${\displaystyle u:X\to Y}$ and${\displaystyle v:Y\to Z}$ are linear maps then${\displaystyle {}^{\text{t}}(v\circ u)={}^{\text{t}}u\circ {}^{\text{t}}v}$^[4]
• If${\displaystyle u:X\to Y}$ is a (surjective) vector space isomorphism then so is the transpose⁠${\displaystyle {}^{\text{t}}u:Y^{\mathrm{\prime }}\to X^{\mathrm{\prime }}}$⁠.
• If${\displaystyle X}$ and${\displaystyle Y}$ arenormed spaces then

$${\displaystyle \Vert x\Vert =\underset{\Vert x^{\mathrm{\prime }}\Vert \le 1}{sup}\left|x^{\mathrm{\prime }}(x)\right|\text{ for each }x\in X}$$and if the linear operator${\displaystyle u:X\to Y}$ is bounded then theoperator norm of${\displaystyle {}^{\text{t}}u}$ is equal to the norm of${\displaystyle u}$; that is^[5][6]$${\displaystyle \Vert u\Vert =\Vert {}^{\text{t}}u\Vert ,}$$and moreover,$${\displaystyle \Vert u\Vert =sup\left\{\left|y^{\mathrm{\prime }}(ux)\right|:\Vert x\Vert \le 1,\Vert y^{\ast }\Vert \le 1\text{ where }x\in X,y^{\mathrm{\prime }}\in Y^{\mathrm{\prime }}\right\}.}$$

Suppose now that${\displaystyle u:X\to Y}$ is a weakly continuous linear operator betweentopological vector spaces${\displaystyle X}$ and${\displaystyle Y}$ with continuous dual spaces${\displaystyle X^{\mathrm{\prime }}}$ and⁠${\displaystyle Y^{\mathrm{\prime }}}$⁠, respectively. Let${\displaystyle ⟨\cdot ,\cdot ⟩:X\times X^{\mathrm{\prime }}\to \mathbb{C}}$ denote the canonicaldual system, defined by${\displaystyle ⟨x,x^{\mathrm{\prime }}⟩=x^{\mathrm{\prime }}x}$ where${\displaystyle x}$ and${\displaystyle x^{\mathrm{\prime }}}$ are said to beorthogonal if⁠${\displaystyle ⟨x,x^{\mathrm{\prime }}⟩=x^{\mathrm{\prime }}x=0}$⁠. For any subsets${\displaystyle A\subseteq X}$ and⁠${\displaystyle S^{\mathrm{\prime }}\subseteq X^{\mathrm{\prime }}}$⁠, let$${\displaystyle A^{\circ }=\left\{x^{\mathrm{\prime }}\in X^{\mathrm{\prime }}:\underset{a\in A}{sup}\left|x^{\mathrm{\prime }}(a)\right|\le 1\right\}\text{ and }{S}^{\circ }=\left\{x\in X:\underset{s^{\mathrm{\prime }}\in S^{\mathrm{\prime }}}{sup}\left|s^{\mathrm{\prime }}(x)\right|\le 1\right\}}$$ denote the (absolute)polar of${\displaystyle A}$ in${\displaystyle X^{\mathrm{\prime }}}$ (resp.of${\displaystyle S^{\mathrm{\prime }}}$ in${\displaystyle X}$).

• If${\displaystyle A\subseteq X}$ and${\displaystyle B\subseteq Y}$ are convex, weakly closed sets containing the origin then${\displaystyle {}^{\text{t}}u\left(B^{\circ }\right)\subseteq A^{\circ }}$ implies⁠${\displaystyle u(A)\subseteq B}$⁠.^[7]
• If${\displaystyle A\subseteq X}$ and${\displaystyle B\subseteq Y}$ then^[4]

$${\displaystyle [u(A){]}^{\circ }={\left({}^{\text{t}}u\right)}^{-1}\left(A^{\circ }\right)}$$and$${\displaystyle u(A)\subseteq B\text{ implies }{}^{\text{t}}u\left(B^{\circ }\right)\subseteq A^{\circ }.}$$

• If${\displaystyle X}$ and${\displaystyle Y}$ arelocally convex then^[5]

$${\displaystyle \ker {}^{\text{t}}u={\left(\mathrm{Im}u\right)}^{\circ }.}$$

Suppose${\displaystyle X}$ and${\displaystyle Y}$ aretopological vector spaces and${\displaystyle u:X\to Y}$ is a weakly continuous linear operator (so⁠${\displaystyle \left({}^{\text{t}}u\right)\left(Y^{\mathrm{\prime }}\right)\subseteq X^{\mathrm{\prime }}}$⁠). Given subsets${\displaystyle M\subseteq X}$ and⁠${\displaystyle N\subseteq X^{\mathrm{\prime }}}$⁠, define theirannihilators (with respect to the canonical dual system) by^[6]

and

• Thekernel of${\displaystyle {}^{\text{t}}u}$ is the subspace of${\displaystyle Y^{\mathrm{\prime }}}$ orthogonal to the image of⁠${\displaystyle u}$⁠:^[7]

$${\displaystyle \ker {}^{\text{t}}u=(\mathrm{Im}u{)}^{\mathrm{\perp }}}$$

• The linear map${\displaystyle u}$ isinjective if and only if its image is a weakly dense subset of${\displaystyle Y}$ (that is, the image of${\displaystyle u}$ is dense in${\displaystyle Y}$ when${\displaystyle Y}$ is given the weak topology induced by⁠${\displaystyle \ker {}^{\text{t}}u}$⁠).^[7]
• The transpose${\displaystyle {}^{\text{t}}u:Y^{\mathrm{\prime }}\to X^{\mathrm{\prime }}}$ is continuous when both${\displaystyle X^{\mathrm{\prime }}}$ and${\displaystyle Y^{\mathrm{\prime }}}$ are endowed with theweak-* topology (resp. both endowed with thestrong 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${\displaystyle X}$ and${\displaystyle Y}$ areFréchet spaces then the continuous linear operator${\displaystyle u:X\to Y}$ issurjective if and only if (1) the transpose${\displaystyle {}^{\text{t}}u:Y^{\mathrm{\prime }}\to X^{\mathrm{\prime }}}$ isinjective, and (2) the image of the transpose of${\displaystyle u}$ is a weakly closed (i.e.weak-* closed) subset of⁠${\displaystyle X^{\mathrm{\prime }}}$⁠.^[9]

Let${\displaystyle M}$ be a closed vector subspace of a Hausdorff locally convex space${\displaystyle X}$ and denote the canonical quotient map by$${\displaystyle \pi :X\to X/M\text{ where }\pi (x):=x+M.}$$Assume${\displaystyle X/M}$ is endowed with thequotient topology induced by the quotient map⁠${\displaystyle \pi :X\to X/M}$⁠. Then the transpose of the quotient map is valued in${\displaystyle M^{\mathrm{\perp }}}$ and$${\displaystyle {}^{\text{t}}\pi :(X/M{)}^{\mathrm{\prime }}\to M^{\mathrm{\perp }}\subseteq X^{\mathrm{\prime }}}$$is a TVS-isomorphism onto⁠${\displaystyle M^{\mathrm{\perp }}}$⁠. If${\displaystyle X}$ is aBanach space then${\displaystyle {}^{\text{t}}\pi :(X/M{)}^{\mathrm{\prime }}\to M^{\mathrm{\perp }}}$ is also anisometry.^[6] Using this transpose, every continuous linear functional on the quotient space${\displaystyle X/M}$ is canonically identified with a continuous linear functional in the annihilator${\displaystyle M^{\mathrm{\perp }}}$ of⁠${\displaystyle M}$⁠.

Let${\displaystyle M}$ be a closed vector subspace of a Hausdorff locally convex space⁠${\displaystyle X}$⁠. If${\displaystyle m^{\mathrm{\prime }}\in M^{\mathrm{\prime }}}$ and if${\displaystyle x^{\mathrm{\prime }}\in X^{\mathrm{\prime }}}$ is a continuous linear extension of${\displaystyle m^{\mathrm{\prime }}}$ to${\displaystyle X}$ then the assignment${\displaystyle m^{\mathrm{\prime }}\mapsto x^{\mathrm{\prime }}+M^{\mathrm{\perp }}}$ induces a vector space isomorphism$${\displaystyle M^{\mathrm{\prime }}\to X^{\mathrm{\prime }}/\left(M^{\mathrm{\perp }}\right),}$$which is an isometry if${\displaystyle X}$ is a Banach space.^[6]

Denote theinclusion map by$${\displaystyle \mathrm{In}:M\to X\text{ where }\mathrm{In}(m):=m\text{ for all }m\in M.}$$The transpose of the inclusion map is$${\displaystyle {}^{\text{t}}\mathrm{In}:X^{\mathrm{\prime }}\to M^{\mathrm{\prime }}}$$whose kernel is the annihilator${\displaystyle M^{\mathrm{\perp }}=\left\{x^{\mathrm{\prime }}\in X^{\mathrm{\prime }}:⟨m,x^{\mathrm{\prime }}⟩=0\text{ for all }m\in M\right\}}$ and which is surjective by theHahn–Banach theorem. This map induces an isomorphism of vector spaces$${\displaystyle X^{\mathrm{\prime }}/\left(M^{\mathrm{\perp }}\right)\to M^{\mathrm{\prime }}.}$$

If the linear map${\displaystyle u}$ is represented by thematrix${\displaystyle A}$ with respect to two bases of${\displaystyle X}$ and⁠${\displaystyle Y}$⁠, then${\displaystyle {}^{\text{t}}u}$ is represented by thetranspose matrix${\displaystyle A^{\text{T}}}$ with respect to the dual bases of${\displaystyle Y^{\mathrm{\prime }}}$ and⁠${\displaystyle X^{\mathrm{\prime }}}$⁠, hence the name. Alternatively, as${\displaystyle u}$ is represented by${\displaystyle A}$ acting to the right on column vectors,${\displaystyle {}^{\text{t}}u}$ 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${\displaystyle {\mathbb{R}}^n}$⁠, which identifies the space of column vectors with the dual space of row vectors.

The identity that characterizes the transpose, that is,⁠${\displaystyle 1}$⁠, is formally similar to the definition of theHermitian adjoint, however, the transpose and the Hermitian adjoint are not the same map. The transpose is a map${\displaystyle Y^{\mathrm{\prime }}\to X^{\mathrm{\prime }}}$ and is defined for linear maps between any vector spaces${\displaystyle X}$ and⁠${\displaystyle Y}$⁠, without requiring any additional structure. The Hermitian adjoint maps${\displaystyle Y\to X}$ and is only defined for linear maps between Hilbert spaces, as it is defined in terms of theinner 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 anondegenerate bilinear form such as the Euclideandot product or anotherrealinner product. In this case, the nondegenerate bilinear form is oftenused implicitly to map between the vector spaces and their duals, to express the transposed map as a map⁠${\displaystyle Y\to X}$⁠. 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${\displaystyle X}$ and${\displaystyle Y}$ are Hilbert spaces and${\displaystyle u:X\to Y}$ is a linear map then the transpose of${\displaystyle u}$ and the Hermitian adjoint of⁠${\displaystyle u}$⁠, which we will denote respectively by${\displaystyle {}^{\text{t}}u}$ and⁠${\displaystyle u^{\ast }}$⁠, are related. Denote by${\displaystyle I:X\to X^{\ast }}$ and${\displaystyle J:Y\to Y^{\ast }}$ the canonical antilinear isometries of the Hilbert spaces${\displaystyle X}$ and${\displaystyle Y}$ onto their duals. Then${\displaystyle u^{\ast }}$ is the following composition of maps:^[10]

${\displaystyle Y\stackrel{J}{⟶}{Y}^{\ast }\stackrel{{}^{\text{t}}u}{⟶}{X}^{\ast }\stackrel{I^{-1}}{⟶}X}$

Suppose that${\displaystyle X}$ and${\displaystyle Y}$ aretopological vector spaces and that${\displaystyle u:X\to Y}$ is a linear map, then many of${\displaystyle u}$'s properties are reflected in⁠${\displaystyle {}^{\text{t}}u}$⁠.

• If${\displaystyle A\subseteq X}$ and${\displaystyle B\subseteq Y}$ are weakly closed, convex sets containing the origin, then${\displaystyle {}^{\text{t}}u(B^{\circ })\subseteq A^{\circ }}$ implies⁠${\displaystyle u(A)\subseteq B}$⁠.^[4]
• The null space of${\displaystyle {}^{\text{t}}u}$ is the subspace of${\displaystyle Y^{\mathrm{\prime }}}$ orthogonal to the range${\displaystyle u(X)}$ of⁠${\displaystyle u}$⁠.^[4]
• ${\displaystyle {}^{\text{t}}u}$ is injective if and only if the range${\displaystyle u(X)}$ of${\displaystyle u}$ 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

• 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.

• Functional analysis
• Linear algebra
• Linear functionals

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