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
Let <math>X^{\#}</math> denote the algebraic dual space of a vector space .
Let <math>X</math> and <math>Y</math> be vector spaces over the same field .
If <math>u : X \to Y</math> is a linear map, then its algebraic adjoint, or dual, is the map <math>{}^{\#\!} u : Y^{\#} \to X^{\#}</math> defined by .
The resulting functional <math>{}^{\#\!} u(f) := f \circ u</math> is called the pullback of <math>f</math> by .
The continuous dual space of a topological vector space (TVS) <math>X</math> is denoted by .
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 , in which case we let <math>{}^\text{t}\! u : Y^{\prime} \to X^{\prime}</math> denote the restriction of <math>{}^{\#\!} u</math> to .
The map <math>{}^\text{t}\! u</math> is called the transpose or algebraic adjoint of .
The following identity characterizes the transpose of :
<math display="block">\left\langle {}^\text{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 .
Properties
The assignment <math>u \mapsto {}^\text{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 .
If <math>X = Y</math> 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 <math>\mathcal{K}</math> to itself.
One can identify <math>{}^\text{t}\!\! \left({}^\text{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>{}^\text{t}\! (v \circ u) = {}^\text{t}\! u \circ {}^\text{t}\! v</math>
- If <math>u : X \to Y</math> is a (surjective) vector space isomorphism then so is the transpose .
- 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 of <math>{}^\text{t}\! u</math> is equal to the norm of <math>u</math>; that is
<math display="block>\|u\| = \left\|{}^\text{t}\! u\right\|,</math>
and moreover,
<math display="block>\|u\| = \sup \left\{\left| y^{\prime}(u x) \right| : \|x\| \leq 1, \left\|y^\prime\right\| \leq 1 \text{ where } x \in X, y^{\prime} \in Y^{\prime} \right\}.</math>
- A bounded linear operator between Banach spaces is compact if and only if its adjoint is compact (Schauder's theorem).
Polars
Suppose now that <math>u : X \to Y</math> is a weakly continuous linear operator between topological vector spaces <math>X</math> and <math>Y</math> with continuous dual spaces <math>X^{\prime}</math> and , 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 if .
For any subsets <math>A \subseteq X</math> and , 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 () (resp. ).
- If <math>A \subseteq X</math> and <math>B \subseteq Y</math> are convex, weakly closed sets containing the origin then <math>{}^\text{t}\! u\left(B^{\circ}\right) \subseteq A^{\circ}</math> implies .
- If <math>A \subseteq X</math> and <math>B \subseteq Y</math> then