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{{Short description|Induced map between the dual spaces of the two vector spaces}}
In [[Linear algebra|linear algebra]], the transpose of a [[Linear map|linear map]] between two vector spaces, defined over the same [[Field (mathematics)|field]], is an induced map between the [[Dual space|dual space]]s 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|adjoint functors]].

== Definition ==

Let <math>X^{\#}</math> denote the algebraic dual space of a vector space <math>X.</math> 
Let <math>X</math> and <math>Y</math> be vector spaces over the same field <math>\mathcal{K}.</math> 
If <math>u : X \to Y</math> is a [[Linear map|linear map]], then its '''algebraic adjoint''' or '''dual''',{{sfn|Schaefer|Wolff|1999|p=128}} is the map <math>{}^{\#} u : Y^{\#} \to X^{\#}</math> defined by <math>f \mapsto f \circ u.</math> 
The resulting functional <math>{}^{\#} u(f) := f \circ u</math> is called the '''[[Pullback|pullback]]''' of <math>f</math> by <math>u.</math> 

The continuous dual space of a [[Topological vector space|topological vector space]] (TVS) <math>X</math> is denoted by <math>X^{\prime}.</math> 
If <math>X</math> and <math>Y</math> are TVSs then a linear map <math>u : X \to Y</math> is '''weakly continuous''' if and only if <math>{}^{\#} u\left(Y^{\prime}\right) \subseteq X^{\prime},</math> in which case we let <math>{}^t u : Y^{\prime} \to X^{\prime}</math> denote the restriction of <math>{}^{\#} u</math> to <math>Y^{\prime}.</math> 
The map <math>{}^t u</math> is called the '''transpose'''{{sfn|Trèves|2006|p=240}} or '''algebraic adjoint''' of <math>u.</math> 
The following identity characterizes the transpose of <math>u</math>:<ref>{{harvtxt|Halmos|1974|loc=§44}}</ref> 
<math display="block">\left\langle {}^t u(f), x \right\rangle = \left\langle f, u(x) \right\rangle \quad \text{ for all } f \in Y ^{\prime} \text{ and } x \in X,</math>
where <math>\left\langle \cdot, \cdot \right\rangle</math> is the natural pairing defined by <math>\left\langle z, h \right\rangle := z(h).</math>

== Properties ==

The assignment <math>u \mapsto {}^t u</math> produces an injective linear map between the space of linear operators from <math>X</math> to <math>Y</math> and the space of linear operators from <math>Y^{\#}</math> to <math>X^{\#}.</math> 
If <math>X = Y</math> then the space of linear maps is an [[Algebra over a field|algebra]] under composition of maps, and the assignment is then an [[Antihomomorphism|antihomomorphism]] of algebras, meaning that <math>{}^t (u v) = {}^t v {}^t u.</math> 
In the language of [[Category theory|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 <math>\mathcal{K}</math> to itself. 
One can identify <math>{}^t \left({}^t u\right)</math> with <math>u</math> using the natural injection into the double dual.
* If <math>u : X \to Y</math> and <math>v : Y \to Z</math> are linear maps then <math>{}^t (v \circ u) = {}^t u \circ {}^t v</math><ref name="Schaefer (1999), pp. 129–130">{{harvnb|Schaefer|Wolff|1999|pp=129–130}}</ref>
* If <math>u : X \to Y</math> is a (surjective) vector space isomorphism then so is the transpose <math>{}^t u : Y^{\prime} \to X^{\prime}.</math>
* If <math>X</math> and <math>Y</math> are normed spaces then 
<math display="block">\|x\| = \sup_{\|x^{\prime}\| \leq 1} \left|x^{\prime}(x) \right| \quad \text{ for each } x \in X</math>
and if the linear operator <math>u : X \to Y</math> is bounded then the [[Operator norm|operator norm]] of <math>{}^t u</math> is equal to the norm of <math>u</math>; that is{{sfn|Trèves|2006|pp=240-252}}{{sfn|Rudin|1991|pp=92-115}}
<math display="block>\|u\| = \left\|{}^t u\right\|,</math>
and moreover,
<math display="block>\|u\| = \sup \left\{\left| y^{\prime}(u x) \right| : \|x\| \leq 1, \left\|y^*\right\| \leq 1 \text{ where } x \in X, y^{\prime} \in Y^{\prime} \right\}.</math>

=== Polars ===

Suppose now that <math>u : X \to Y</math> is a weakly continuous linear operator between [[Topological vector space|topological vector space]]s <math>X</math> and <math>Y</math> with continuous dual spaces <math>X^{\prime}</math> and <math>Y^{\prime},</math> respectively. 
Let <math>\langle \cdot, \cdot \rangle : X \times X^{\prime} \to \Complex</math> denote the canonical [[Dual system|dual system]], defined by <math>\left\langle x, x^{\prime} \right\rangle = x^{\prime} x</math> where <math>x</math> and <math>x^{\prime}</math> are said to be {{em|orthogonal}} if <math>\left\langle x, x^{\prime} \right\rangle = x^{\prime} x = 0.</math> 
For any subsets <math>A \subseteq X</math> and <math>S^{\prime} \subseteq X^{\prime},</math> let 
<math display="block">A^{\circ} = \left\{ x^{\prime} \in X^{\prime} : \sup_{a \in A} \left|x^{\prime}(a)\right| \leq 1 \right\} \qquad \text{ and } \qquad S^{\circ} = \left\{ x \in X : \sup_{s^{\prime} \in S^{\prime}} \left|s^{\prime}(x)\right| \leq 1 \right\}</math> 
denote the ({{em|absolute}}) {{em|[[Polar set|polar]] of <math>A</math> in <math>X^{\prime}</math>}} (resp. {{em|of <math>S^{\prime}</math> in <math>X</math>}}). 

* If <math>A \subseteq X</math> and <math>B \subseteq Y</math> are convex, weakly closed sets containing the origin then <math>{}^t u\left(B^{\circ}\right) \subseteq A^{\circ}</math> implies <math>u(A) \subseteq B.</math>{{sfn|Schaefer|Wolff|1999|pp=128–130}}
* If <math>A \subseteq X</math> and <math>B \subseteq Y</math> then<ref name="Schaefer (1999), pp. 129–130"/>
<math display="block">[u(A)]^{\circ} = \left({}^t u\right)^{-1}\left(A^{\circ}\right)</math>
and
<math display="block">u(A) \subseteq B \quad \text{ implies } \quad {}^t u\left(B^{\circ}\right) \subseteq A^{\circ}.</math>
* If <math>X</math> and <math>Y</math> are [[Locally convex topological vector space|locally convex]] then{{sfn|Trèves|2006|pp=240-252}} 
<math display="block">\operatorname{ker} {}^t u = \left(\operatorname{Im} u\right)^{\circ}.</math>

=== Annihilators ===

Suppose <math>X</math> and <math>Y</math> are [[Topological vector space|topological vector space]]s and <math>u : X \to Y</math> is a weakly continuous linear operator (so <math>\left({}^t u\right)\left(Y^{\prime}\right) \subseteq X^{\prime}</math>). Given subsets <math>M \subseteq X</math> and <math>N \subseteq X^{\prime},</math> define their {{em|[[Dual space#Quotient spaces and annihilators|annihilators]]}} (with respect to the canonical dual system) by{{sfn|Rudin|1991|pp=92-115}}
:<math>\begin{alignat}{4}
M^{\bot} :&= \left\{ x^{\prime} \in X^{\prime} : \left\langle m, x^{\prime} \right\rangle = 0 \text{ for all } m \in M \right\} \\
&= \left\{ x^{\prime} \in X^{\prime} : x^{\prime}(M) = \{0\} \right\} \qquad \text{ where } x^{\prime}(M) := \left\{ x^{\prime}(m) : m \in M \right\}
\end{alignat}</math>
and
:<math>\begin{alignat}{4}
{}^{\bot} N :&= \left\{ x \in X : \left\langle x, n^{\prime} \right\rangle = 0 \text{ for all } n^{\prime} \in N \right\} \\
&= \left\{ x \in X : N(x) = \{ 0 \} \right\} \qquad \text{ where } N(x) := \left\{ n^{\prime}(x) : n^{\prime} \in N \right\} \\
\end{alignat}</math>

* The [[Kernel (linear algebra)|kernel]] of <math>{}^t u</math> is the subspace of <math>Y^{\prime}</math> orthogonal to the image of <math>u</math>:{{sfn|Schaefer|Wolff|1999|pp=128–130}}
<math display="block">\ker {}^t u = (\operatorname{Im} u)^{\bot}</math>
* The linear map <math>u</math> is injective if and only if its image is a weakly dense subset of <math>Y</math> (that is, the image of <math>u</math> is dense in <math>Y</math> when <math>Y</math> is given the weak topology induced by <math>\operatorname{ker} {}^t u</math>).{{sfn|Schaefer|Wolff|1999|pp=128–130}}
* The transpose <math>{}^t u : Y^{\prime} \to X^{\prime}</math> is continuous when both <math>X^{\prime}</math> and <math>Y^{\prime}</math> 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).{{sfn|Trèves|2006|pp=199-200}}
* ([[Surjection of Fréchet spaces]]): If <math>X</math> and <math>Y</math> are [[Fréchet space]]s then the continuous linear operator <math>u : X \to Y</math> is surjective if and only if (1) the transpose <math>{}^t u : Y^{\prime} \to X^{\prime}</math> is injective, and (2) the image of the transpose of <math>u</math> is a weakly closed (i.e. weak-* closed) subset of <math>X^{\prime}.</math>{{sfn|Trèves|2006|pp=382-383}}

=== Duals of quotient spaces ===

Let <math>M</math> be a closed vector subspace of a Hausdorff locally convex space <math>X</math> and denote the canonical quotient map by 
<math display="block">\pi : X \to X / M \quad \text{ where } \quad \pi(x) := x + M.</math>
Assume <math>X / M</math> is endowed with the quotient topology induced by the quotient map <math>\pi : X \to X / M.</math> 
Then the transpose of the quotient map is valued in <math>M^{\bot}</math> and
<math display="block">{}^t \pi : (X / M)^{\prime} \to M^{\bot} \subseteq X^{\prime}</math>
is a TVS-isomorphism onto <math>M^{\bot}.</math> 
If <math>X</math> is a Banach space then <math>{}^t \pi : (X / M)^{\prime} \to M^{\bot}</math> is also an [[Isometry|isometry]].{{sfn|Rudin|1991|pp=92-115}} 
Using this transpose, every continuous linear functional on the quotient space <math>X / M</math> is canonically identified with a continuous linear functional in the annihilator <math>M^{\bot}</math> of <math>M.</math>

=== Duals of vector subspaces ===

Let <math>M</math> be a closed vector subspace of a Hausdorff locally convex space <math>X.</math> 
If <math>m^{\prime} \in M^{\prime}</math> and if <math>x^{\prime} \in X^{\prime}</math> is a continuous linear extension of <math>m^{\prime}</math> to <math>X</math> then the assignment <math>m^{\prime} \mapsto x^{\prime} + M^{\bot}</math> induces a vector space isomorphism 
<math display="block">M^{\prime} \to X^{\prime} / \left(M^{\bot}\right),</math>
which is an isometry if <math>X</math> is a Banach space.{{sfn|Rudin|1991|pp=92-115}} 

Denote the [[Inclusion map|inclusion map]] by
<math display="block">\operatorname{In} : M \to X \quad \text{ where } \quad \operatorname{In}(m) := m \quad \text{ for all } m \in M.</math>
The transpose of the inclusion map is 
<math display="block">{}^t \operatorname{In} : X^{\prime} \to M^{\prime}</math>
whose kernel is the annihilator <math>M^{\bot} = \left\{ x^{\prime} \in X^{\prime} : \left\langle m, x^{\prime} \right\rangle = 0 \text{ for all } m \in M \right\}</math> and which is surjective by the [[Hahn–Banach theorem]]. This map induces an isomorphism of vector spaces
<math display="block">X^{\prime} / \left(M^{\bot}\right) \to M^{\prime}.</math>

== Representation as a matrix ==

If the linear map <math>u</math> is represented by the [[Matrix (mathematics)|matrix]] <math>A</math> with respect to two bases of <math>X</math> and <math>Y,</math> then <math>{}^t u</math> is represented by the [[Transpose|transpose]] matrix <math>A^T</math> with respect to the dual bases of <math>Y^{\prime}</math> and <math>X^{\prime},</math> hence the name. 
Alternatively, as <math>u</math> is represented by <math>A</math> acting to the right on column vectors, <math>{}^t u</math> 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 <math>\R^n,</math> which identifies the space of column vectors with the dual space of row vectors.

== Relation to the Hermitian adjoint ==

{{Main|Hermitian adjoint}}

The identity that characterizes the transpose, that is, <math>\left[u^{*}(f), x\right] = [f, u(x)],</math> 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 <math>Y^{\prime} \to X^{\prime}</math> and is defined for linear maps between any vector spaces <math>X</math> and <math>Y,</math> without requiring any additional structure. 
The Hermitian adjoint maps <math>Y \to X</math> 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|dot product]] or another {{em|real}} inner product. 
In this case, the nondegenerate bilinear form is often [[Dual space#Bilinear products and dual spaces|used]] implicitly to map between the vector spaces and their duals, to express the transposed map as a map <math>Y \to X.</math> 
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 <math>X</math> and <math>Y</math> are Hilbert spaces and <math>u : X \to Y</math> is a linear map then the transpose of <math>u</math> and the Hermitian adjoint of <math>u,</math> which we will denote respectively by <math>{}^t u</math> and <math>u^{*},</math> are related. 
Denote by <math>I : X \to X^{*}</math> and <math>J : Y \to Y^{*}</math> the canonical antilinear isometries of the Hilbert spaces <math>X</math> and <math>Y</math> onto their duals. 
Then <math>u^{*}</math> is the following composition of maps:{{sfn|Trèves|2006|p=488}}

:<math>Y \overset{J}{\longrightarrow} Y^* \overset{{}^{\text{t}}u}{\longrightarrow} X^* \overset{I^{-1}}{\longrightarrow} X</math>

== Applications to functional analysis ==

Suppose that <math>X</math> and <math>Y</math> are [[Topological vector space|topological vector space]]s and that <math>u : X \to Y</math> is a linear map, then many of <math>u</math>'s properties are reflected in <math>{}^t u.</math>

* If <math>A \subseteq X</math> and <math>B \subseteq Y</math> are weakly closed, convex sets containing the origin, then <math>{}^t u\left(B^{\circ}\right) \subseteq A^{\circ}</math> implies <math>u(A) \subseteq B.</math><ref name="Schaefer (1999), pp. 129–130" />
* The null space of <math>{}^t u</math> is the subspace of <math>Y^{\prime}</math> orthogonal to the range <math>u(X)</math> of <math>u.</math><ref name="Schaefer (1999), pp. 129–130" />
* <math>{}^t u</math> is injective if and only if the range <math>u(X)</math> of <math>u</math> is weakly closed.<ref name="Schaefer (1999), pp. 129–130" />

== See also ==

* {{annotated link|Adjoint functors}}
* {{annotated link|Composition operator}}
* {{annotated link|Hermitian adjoint}}
* {{annotated link|Riesz representation theorem}}
* {{section link|Dual space|Transpose of a linear map}}
* {{section link|Transpose|Transpose of a linear map}}

== References ==

{{reflist|group=note}}
{{reflist}}

== Bibliography ==

* {{citation|first=Paul|last=Halmos|title=Finite-dimensional Vector Spaces|year=1974|publisher=Springer|isbn=0-387-90093-4}}
* {{Rudin Walter Functional Analysis|edition=2}} <!-- {{sfn|Rudin |1991|p=}} -->
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn|Schaefer|Wolff|1999|p=}} -->
* {{Trèves François Topological vector spaces, distributions and kernels}} <!-- {{sfn|Trèves|2006|p=}} -->

{{linear algebra}}
{{Functional analysis}}

[[Category:Functional analysis]]
[[Category:Linear algebra]]

{{Sourceattribution|Transpose of a linear map}}