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{{Short description|Induced map between the dual spaces of the two vector spaces}}
{{See also|Transpose|Dual system#Transposes|Transpose#Transposes of linear maps and bilinear forms}}
In [[linear algebra]], the transpose of a [[linear map]] between two vector spaces, defined over the same [[Field (mathematics)|field]], is an induced map between the [[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]].

== Definition ==
{{See also|Dual system#Transposes|Transpose#Transposes of linear maps and bilinear forms}}

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]], 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]]''' of <math>f</math> by <math>u.</math> 

The [[continuous dual space]] of a [[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]] of algebras, meaning that <math>{}^t (u v) = {}^t v {}^t u.</math> 
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 <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 space]]s 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]] 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]]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]], 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 vectors|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]]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-* topology|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]].{{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]] 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]] 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}}
{{See also|Riesz representation theorem}}

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 [[Adjoint of an operator|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 form|nondegenerate bilinear form]] such as the Euclidean [[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]]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|authorlink=Paul Halmos|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}}
{{Duality and spaces of linear maps}}
{{Functional analysis}}

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