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In
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. ...
, the transpose of a
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
between two vector spaces, defined over the same
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
, is an induced map between the
dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...
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 In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kno ...
.


Definition

Let X^ denote the
algebraic dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by con ...
of a vector space X. Let X and Y be vector spaces over the same field \mathcal. If u : X \to Y is a
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
, then its algebraic adjoint or dual, is the map ^ u : Y^ \to X^ defined by f \mapsto f \circ u. The resulting functional ^ u(f) := f \circ u is called the
pullback In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward. Precomposition Precomposition with a function probably provides the most elementary notion of pullback: i ...
of f by u. The
continuous dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by cons ...
of a
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
(TVS) X is denoted by X^. If X and Y are TVSs then a linear map u : X \to Y is weakly continuous if and only if ^ u\left(Y^\right) \subseteq X^, in which case we let ^t u : Y^ \to X^ denote the restriction of ^ u to Y^. The map ^t u is called the transpose or algebraic adjoint of u. The following identity characterizes the transpose of u \left\langle ^t u(f), x \right\rangle = \left\langle f, u(x) \right\rangle \quad \text f \in Y ^ \text x \in X^ where \left\langle \cdot, \cdot \right\rangle is the natural pairing defined by \left\langle z, h \right\rangle := z(h).


Properties

The assignment u \mapsto ^t u produces an
injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
linear map between the space of linear operators from X to Y and the space of linear operators from Y^ to X^. If X = Y then the space of linear maps is an
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
under
composition of maps In mathematics, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and ...
, and the assignment is then an
antihomomorphism In mathematics, an antihomomorphism is a type of function defined on sets with multiplication that reverses the order of multiplication. An antiautomorphism is a bijective antihomomorphism, i.e. an antiisomorphism, from a set to itself. From ...
of algebras, meaning that ^t (u v) = ^t v ^t u. In the language of
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
, taking the dual of vector spaces and the transpose of linear maps is therefore a
contravariant functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ma ...
from the category of vector spaces over \mathcal to itself. One can identify ^t \left(^t u\right) with u using the natural injection into the double dual. * If u : X \to Y and v : Y \to Z are linear maps then ^t (v \circ u) = ^t u \circ ^t v * If u : X \to Y is a (
surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
) vector space isomorphism then so is the transpose ^t u : Y^ \to X^. * If X and Y are
normed space In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" i ...
s then \, x\, = \sup_ \left, x^(x) \ \quad \text x \in X and if the linear operator u : X \to Y is bounded then the
operator norm In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its . Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces. Introdu ...
of ^t u is equal to the norm of u; that is \, u\, = \sup \left\.


Polars

Suppose now that u : X \to Y is a weakly continuous linear operator between
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
s X and Y with continuous dual spaces X^ and Y^, respectively. Let \langle \cdot, \cdot \rangle : X \times X^ \to \Complex denote the canonical
dual system In mathematics, a dual system, dual pair, or duality over a field \mathbb is a triple (X, Y, b) consisting of two vector spaces X and Y over \mathbb and a non- degenerate bilinear map b : X \times Y \to \mathbb. Duality theory, the study of dual ...
, defined by \left\langle x, x^ \right\rangle = x^ x where x and x^ are said to be if \left\langle x, x^ \right\rangle = x^ x = 0. For any subsets A \subseteq X and S^ \subseteq X^, let A^ = \left\ \qquad \text \qquad S^ = \left\ denote the () (resp. ). * If A \subseteq X and B \subseteq Y are convex, weakly closed sets containing the origin then ^t u\left(B^\right) \subseteq A^ implies u(A) \subseteq B. * If A \subseteq X and B \subseteq Y then (A) = \left(^t u\right)^\left(A^\right) and u(A) \subseteq B \quad \text \quad ^t u\left(B^\right) \subseteq A^. * If X and Y are locally convex then \operatorname ^t u = \left(\operatorname u\right)^.


Annihilators

Suppose X and Y are
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
s and u : X \to Y is a weakly continuous linear operator (so \left(^t u\right)\left(Y^\right) \subseteq X^). Given subsets M \subseteq X and N \subseteq X^, define their (with respect to the canonical dual system) by :\begin M^ :&= \left\ \\ &= \left\ \qquad \text x^(M) := \left\ \end and :\begin ^ N :&= \left\ \\ &= \left\ \qquad \text N(x) := \left\ \\ \end * The
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learnin ...
of ^t u is the subspace of Y^ orthogonal to the image of u: \ker ^t u = (\operatorname u)^ * The linear map u is
injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
if and only if its image is a weakly dense subset of Y (that is, the image of u is dense in Y when Y is given the weak topology induced by \operatorname ^t u). * The transpose ^t u : Y^ \to X^ is continuous when both X^ and Y^ are endowed with the
weak-* topology In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...
(resp. both endowed with the
strong dual In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) X is the continuous dual space X^ of X equipped with the strong (dual) topology or the topology of uniform convergence on bounded sub ...
topology, both endowed with the topology of uniform convergence on compact convex subsets, both endowed with the topology of uniform convergence on compact subsets). * (
Surjection of Fréchet spaces The theorem on the surjection of Fréchet spaces is an important theorem, due to Stefan Banach, that characterizes when a continuous linear operator between Fréchet spaces is surjective. The importance of this theorem is related to the open m ...
): If X and Y are
Fréchet space In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces ( normed vector spaces that are complete with respect to th ...
s then the continuous linear operator u : X \to Y is
surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
if and only if (1) the transpose ^t u : Y^ \to X^ is
injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
, and (2) the image of the transpose of u is a weakly closed (i.e. weak-* closed) subset of X^.


Duals of quotient spaces

Let M be a closed vector subspace of a Hausdorff locally convex space X and denote the canonical quotient map by \pi : X \to X / M \quad \text \quad \pi(x) := x + M. Assume X / M is endowed with the
quotient topology In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient t ...
induced by the quotient map \pi : X \to X / M. Then the transpose of the quotient map is valued in M^ and ^t \pi : (X / M)^ \to M^ \subseteq X^ is a TVS-isomorphism onto M^. If X is a Banach space then ^t \pi : (X / M)^ \to M^ is also an
isometry In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
. Using this transpose, every continuous linear functional on the quotient space X / M is canonically identified with a continuous linear functional in the annihilator M^ of M.


Duals of vector subspaces

Let M be a closed vector subspace of a Hausdorff locally convex space X. If m^ \in M^ and if x^ \in X^ is a continuous linear extension of m^ to X then the assignment m^ \mapsto x^ + M^ induces a vector space isomorphism M^ \to X^ / \left(M^\right), which is an isometry if X is a Banach space. Denote the
inclusion map In mathematics, if A is a subset of B, then the inclusion map (also inclusion function, insertion, or canonical injection) is the function \iota that sends each element x of A to x, treated as an element of B: \iota : A\rightarrow B, \qquad \iot ...
by \operatorname : M \to X \quad \text \quad \operatorname(m) := m \quad \text m \in M. The transpose of the inclusion map is ^t \operatorname : X^ \to M^ whose kernel is the annihilator M^ = \left\ and which is surjective by the
Hahn–Banach theorem The Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear f ...
. This map induces an isomorphism of vector spaces X^ / \left(M^\right) \to M^.


Representation as a matrix

If the linear map u is represented by the
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
A with respect to two bases of X and Y, then ^t u is represented by the
transpose In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The tr ...
matrix A^T with respect to the dual bases of Y^ and X^, hence the name. Alternatively, as u is represented by A acting to the right on column vectors, ^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 \R^n, which identifies the space of column vectors with the dual space of row vectors.


Relation to the Hermitian adjoint

The identity that characterizes the transpose, that is, \left ^(f), x\right= , u(x) is formally similar to the definition of the
Hermitian adjoint In mathematics, specifically in operator theory, each linear operator A on a Euclidean vector space defines a Hermitian adjoint (or adjoint) operator A^* on that space according to the rule :\langle Ax,y \rangle = \langle x,A^*y \rangle, where ...
, however, the transpose and the Hermitian adjoint are not the same map. The transpose is a map Y^ \to X^ and is defined for linear maps between any vector spaces X and Y, without requiring any additional structure. The Hermitian adjoint maps Y \to X and is only defined for linear maps between Hilbert spaces, as it is defined in terms of the
inner product In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
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 In mathematics, specifically linear algebra, a degenerate bilinear form on a vector space ''V'' is a bilinear form such that the map from ''V'' to ''V''∗ (the dual space of ''V'' ) given by is not an isomorphism. An equivalent defin ...
such as the Euclidean
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
or another
inner product In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
. In this case, the nondegenerate bilinear form is often
used Used may refer to: Common meanings *Used good, goods of any type that have been used before or pre-owned *Used to, English auxiliary verb Places *Used, Huesca, a village in Huesca, Aragon, Spain *Used, Zaragoza, a town in Zaragoza, Aragon, Spain ...
implicitly to map between the vector spaces and their duals, to express the transposed map as a map 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 X and Y are Hilbert spaces and u : X \to Y is a linear map then the transpose of u and the Hermitian adjoint of u, which we will denote respectively by ^t u and u^, are related. Denote by I : X \to X^ and J : Y \to Y^ the canonical antilinear isometries of the Hilbert spaces X and Y onto their duals. Then u^ is the following composition of maps: :Y \overset Y^* \overset X^* \overset X


Applications to functional analysis

Suppose that X and Y are
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
s and that u : X \to Y is a linear map, then many of u's properties are reflected in ^t u. * If A \subseteq X and B \subseteq Y are weakly closed, convex sets containing the origin, then ^t u\left(B^\right) \subseteq A^ implies u(A) \subseteq B. * The null space of ^t u is the subspace of Y^ orthogonal to the range u(X) of u. * ^t u is injective if and only if the range u(X) of u is weakly closed.


See also

* * * * * *


References


Bibliography

* * * * {{Functional analysis Functional analysis Linear algebra