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In mathematics, vector space of linear forms

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Inmathematics, anyvector space $V {\displaystyle V}$ has a correspondingdual vector space (or justdual space for short) consisting of alllinear forms on $V, {\displaystyle V,}$ together with the vector space structure ofpointwise addition andscalar multiplication by constants.

The dual space as defined above is defined for all vector spaces, and to avoid ambiguity may also be called thealgebraic dual space.When defined for atopological vector space, there is a subspace of the dual space, corresponding tocontinuous linear functionals, called thecontinuous dual space.

Dual vector spaces find application in many branches of mathematics that use vector spaces, such as intensor analysis withfinite-dimensional vector spaces.When applied to vector spaces of functions (which are typically infinite-dimensional), dual spaces are used to describemeasures,distributions, andHilbert spaces. Consequently, the dual space is an important concept infunctional analysis.

Early terms fordual includepolarer Raum [Hahn 1927],espace conjugué,adjoint space [Alaoglu 1940], andtransponierter Raum [Schauder 1930] and [Banach 1932]. The termdual is due toBourbaki 1938.^[1]

Algebraic dual space

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Given anyvector space $V {\displaystyle V}$ over afield $F {\displaystyle F}$ , the(algebraic) dual space $V^{*}$ ^[2] (alternatively denoted by $V^{\lor }$ ^[3] or $V^{'} {\displaystyle V'}$ ^[4]^[5])^{[nb 1]} is defined as the set of alllinear maps $\varphi :V\to F$ (linear functionals). Since linear maps are vector spacehomomorphisms, the dual space may be denoted $\hom(V,F)$ .^[3]The dual space $V^{*}$ itself becomes a vector space over $F {\displaystyle F}$ when equipped with an addition and scalar multiplication satisfying:

{\begin{aligned}(\varphi +\psi )(x)&=\varphi (x)+\psi (x)\\(a\varphi )(x)&=a\left(\varphi (x)\right)\end{aligned}}

for all $\varphi ,\psi \in V^{*}$ , $x\in V$ , and $a\in F$ .

Elements of the algebraic dual space $V^{*}$ are sometimes calledcovectors,one-forms, orlinear forms.

The pairing of a functional $\varphi$ in the dual space $V^{*}$ and an element $x {\displaystyle x}$ of $V {\displaystyle V}$ is sometimes denoted by a bracket: $\varphi (x)=[x,\varphi ]$ ^[6]or $\varphi (x)=\langle x,\varphi \rangle$ .^[7] This pairing defines a nondegeneratebilinear mapping^{[nb 2]} $\langle \cdot ,\cdot \rangle :V\times V^{*}\to F$ called thenatural pairing.

Finite-dimensional case

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Infinite-dimensional case

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If $V {\displaystyle V}$ is not finite-dimensional but has abasis^{[nb 3]} $\mathbf {e} _{\alpha }$ indexed by an infinite set $A {\displaystyle A}$ , then the same construction as in the finite-dimensional case yieldslinearly independent elements $\mathbf {e} ^{\alpha }$ ( $\alpha \in A$ ) of the dual space, but they will not form a basis.

For instance, consider the space $\mathbb {R} ^{\infty }$ , whose elements are thosesequences of real numbers that contain only finitely many non-zero entries, which has a basis indexed by the natural numbers $\mathbb {N}$ . For $i\in \mathbb {N}$ , $\mathbf {e} _{i}$ is the sequence consisting of all zeroes except in the $i {\displaystyle i}$ -th position, which is 1.The dual space of $\mathbb {R} ^{\infty }$ is (isomorphic to) $\mathbb {R} ^{\mathbb {N} }$ , the space ofall sequences of real numbers: each real sequence $(a_{n})$ defines a function where the element $(x_{n})$ of $\mathbb {R} ^{\infty }$ is sent to the number

\sum _{n}a_{n}x_{n},

which is a finite sum because there are only finitely many nonzero $x_{n}$ . Thedimension of $\mathbb {R} ^{\infty }$ iscountably infinite, whereas $\mathbb {R} ^{\mathbb {N} }$ does not have a countable basis.

This observation generalizes to any^{[nb 3]} infinite-dimensional vector space $V {\displaystyle V}$ over any field $F {\displaystyle F}$ : a choice of basis $\{\mathbf {e} _{\alpha }:\alpha \in A\}$ identifies $V {\displaystyle V}$ with the space $(F^{A})_{0}$ of functions $f:A\to F$ such that $f_{\alpha }=f(\alpha )$ is nonzero for only finitely many $\alpha \in A$ , where such a function $f {\displaystyle f}$ is identified with the vector

\sum _{\alpha \in A}f_{\alpha }\mathbf {e} _{\alpha }

in $V {\displaystyle V}$ (the sum is finite by the assumption on $f {\displaystyle f}$ , and any $v\in V$ may be written uniquely in this way by the definition of the basis).

The dual space of $V {\displaystyle V}$ may then be identified with the space $F^{A}$ ofall functions from $A {\displaystyle A}$ to $F {\displaystyle F}$ : a linear functional $T {\displaystyle T}$ on $V {\displaystyle V}$ is uniquely determined by the values $\theta _{\alpha }=T(\mathbf {e} _{\alpha })$ it takes on the basis of $V {\displaystyle V}$ , and any function $\theta :A\to F$ (with $\theta (\alpha )=\theta _{\alpha }$ ) defines a linear functional $T {\displaystyle T}$ on $V {\displaystyle V}$ by

T\left(\sum _{\alpha \in A}f_{\alpha }\mathbf {e} _{\alpha }\right)=\sum _{\alpha \in A}f_{\alpha }T(e_{\alpha })=\sum _{\alpha \in A}f_{\alpha }\theta _{\alpha }.

Again, the sum is finite because $f_{\alpha }$ is nonzero for only finitely many $\alpha$ .

The set $(F^{A})_{0}$ may be identified (essentially by definition) with thedirect sum of infinitely many copies of $F {\displaystyle F}$ (viewed as a 1-dimensional vector space over itself) indexed by $A {\displaystyle A}$ , i.e. there are linear isomorphisms

V\cong (F^{A})_{0}\cong \bigoplus _{\alpha \in A}F.

On the other hand, $F^{A}$ is (again by definition), thedirect product of infinitely many copies of $F {\displaystyle F}$ indexed by $A {\displaystyle A}$ , and so the identification

V^{*}\cong \left(\bigoplus _{\alpha \in A}F\right)^{*}\cong \prod _{\alpha \in A}F^{*}\cong \prod _{\alpha \in A}F\cong F^{A}

is a special case of ageneral result relating direct sums (ofmodules) to direct products.

If a vector space is not finite-dimensional, then its (algebraic) dual space isalways of larger dimension (as acardinal number) than the original vector space. This is in contrast to the case of the continuous dual space, discussed below, which may beisomorphic to the original vector space even if the latter is infinite-dimensional.

The proof of this inequality between dimensions results from the following.

If $V {\displaystyle V}$ is an infinite-dimensional $F {\displaystyle F}$ -vector space, the arithmetical properties ofcardinal numbers implies that

\mathrm {dim} (V)=|A|<|F|^{|A|}=|V^{\ast }|=\mathrm {max} (|\mathrm {dim} (V^{\ast })|,|F|),

where cardinalities are denoted asabsolute values. For proving that $\mathrm {dim} (V)<\mathrm {dim} (V^{*}),$ it suffices to prove that $|F|\leq |\mathrm {dim} (V^{\ast })|,$ which can be done with an argument similar toCantor's diagonal argument.^[9] The exact dimension of the dual is given by theErdős–Kaplansky theorem.

Bilinear products and dual spaces

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IfV is finite-dimensional, thenV is isomorphic toV^∗. But there is in general nonatural isomorphism between these two spaces.^[10] Anybilinear form $\langle \cdot ,\cdot \rangle$ on $V {\displaystyle V}$ gives a mapping of $V {\displaystyle V}$ into its dual space via

v\mapsto \langle v,\cdot \rangle

where the right hand side is defined as the functional onV taking each $w\in V$ to $\langle v,w\rangle$ . In other words, the bilinear form determines a linear mapping

\Phi _{\langle \cdot ,\cdot \rangle }:V\to V^{*}

defined by

\left[\Phi _{\langle \cdot ,\cdot \rangle }(v),w\right]=\langle v,w\rangle .

If the bilinear form isnondegenerate, then this is an isomorphism onto a subspace ofV^∗.IfV is finite-dimensional, then this is an isomorphism onto all ofV^∗. Conversely, any isomorphism $\Phi$ fromV to a subspace ofV^∗ (resp., all ofV^∗ ifV is finite dimensional) defines a unique nondegenerate bilinear form $\langle \cdot ,\cdot \rangle _{\Phi }$ onV by

\langle v,w\rangle _{\Phi }=(\Phi (v))(w)=[\Phi (v),w].\,

Thus there is a one-to-one correspondence between isomorphisms ofV to a subspace of (resp., all of)V^∗ and nondegenerate bilinear forms onV.

If the vector spaceV is over thecomplex field, then sometimes it is more natural to considersesquilinear forms instead of bilinear forms.In that case, a given sesquilinear form⟨·,·⟩ determines an isomorphism ofV with thecomplex conjugate of the dual space

\Phi _{\langle \cdot ,\cdot \rangle }:V\to {\overline {V^{*}}}.

The conjugate of the dual space ${\overline {V^{*}}}$ can be identified with the set of all additive complex-valued functionalsf :V →C such that

f(\alpha v)={\overline {\alpha }}f(v).

Injection into the double-dual

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There is anatural homomorphism $\Psi$ from $V {\displaystyle V}$ into the double dual $V^{**}=\hom(V^{*},F)$ , defined by $(\Psi (v))(\varphi )=\varphi (v)$ for all $v\in V,\varphi \in V^{*}$ . In other words, if $\mathrm {ev} _{v}:V^{*}\to F$ is the evaluation map defined by $\varphi \mapsto \varphi (v)$ , then $\Psi :V\to V^{**}$ is defined as the map $v\mapsto \mathrm {ev} _{v}$ . This map $\Psi$ is alwaysinjective;^{[nb 3]} and it is always anisomorphism if $V {\displaystyle V}$ is finite-dimensional.^[11]Indeed, the isomorphism of a finite-dimensional vector space with its double dual is an archetypal example of anatural isomorphism.Infinite-dimensional Hilbert spaces are not isomorphic to their algebraic double duals, but instead to their continuous double duals.

Transpose of a linear map

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Main article:Transpose of a linear map

Iff :V →W is alinear map, then thetranspose (ordual)f^∗ :W^∗ →V^∗ is defined by

f^{*}(\varphi )=\varphi \circ f\,

for every $\varphi \in W^{*}$ . The resulting functional $f^{*}(\varphi )$ in $V^{*}$ is called thepullback of $\varphi$ along $f {\displaystyle f}$ .

The following identity holds for all $\varphi \in W^{*}$ and $v\in V$ :

[f^{*}(\varphi ),\,v]=[\varphi ,\,f(v)],

where the bracket [·,·] on the left is the natural pairing ofV with its dual space, and that on the right is the natural pairing ofW with its dual. This identity characterizes the transpose,^[12] and is formally similar to the definition of theadjoint.

The assignmentf ↦f^∗ produces aninjective linear map between the space of linear operators fromV toW and the space of linear operators fromW^∗ toV^∗; this homomorphism is anisomorphism if and only ifW is finite-dimensional.IfV =W then the space of linear maps is actually analgebra undercomposition of maps, and the assignment is then anantihomomorphism of algebras, meaning that(fg)^∗ =g^∗f^∗.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 overF to itself.It is possible to identify (f^∗)^∗ withf using the natural injection into the double dual.

If the linear mapf is represented by thematrixA with respect to two bases ofV andW, thenf^∗ is represented by thetranspose matrixA^T with respect to the dual bases ofW^∗ andV^∗, hence the name.Alternatively, asf is represented byA acting on the left on column vectors,f^∗ is represented by the same matrix acting on the right on row vectors.These points of view are related by the canonical inner product onRⁿ, which identifies the space of column vectors with the dual space of row vectors.

Quotient spaces and annihilators

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Let $S {\displaystyle S}$ be a subset of $V {\displaystyle V}$ .Theannihilator of $S {\displaystyle S}$ in $V^{*}$ , denoted here $S^{0}$ , is the collection of linear functionals $f\in V^{*}$ such that $[f,s]=0$ for all $s\in S$ .That is, $S^{0}$ consists of all linear functionals $f:V\to F$ such that the restriction to $S {\displaystyle S}$ vanishes: $f|_{S}=0$ .Within finite dimensional vector spaces, the annihilator is dual to (isomorphic to) theorthogonal complement.

The annihilator of a subset is itself a vector space.The annihilator of the zero vector is the whole dual space: $\{0\}^{0}=V^{*}$ , and the annihilator of the whole space is just the zero covector: $V^{0}=\{0\}\subseteq V^{*}$ .Furthermore, the assignment of an annihilator to a subset of $V {\displaystyle V}$ reverses inclusions, so that if $\{0\}\subseteq S\subseteq T\subseteq V$ , then

\{0\}\subseteq T^{0}\subseteq S^{0}\subseteq V^{*}.

If $A {\displaystyle A}$ and $B {\displaystyle B}$ are two subsets of $V {\displaystyle V}$ then

A^{0}+B^{0}\subseteq (A\cap B)^{0}.

If $(A_{i})_{i\in I}$ is any family of subsets of $V {\displaystyle V}$ indexed by $i {\displaystyle i}$ belonging to some index set $I {\displaystyle I}$ , then

\left(\bigcup _{i\in I}A_{i}\right)^{0}=\bigcap _{i\in I}A_{i}^{0}.

In particular if $A {\displaystyle A}$ and $B {\displaystyle B}$ are subspaces of $V {\displaystyle V}$ then

(A+B)^{0}=A^{0}\cap B^{0}

and^{[nb 3]}

(A\cap B)^{0}=A^{0}+B^{0}.

If $V {\displaystyle V}$ is finite-dimensional and $W {\displaystyle W}$ is avector subspace, then

W^{00}=W

after identifying $W {\displaystyle W}$ with its image in the second dual space under the double duality isomorphism $V\approx V^{**}$ . In particular, forming the annihilator is aGalois connection on the lattice of subsets of a finite-dimensional vector space.

If $W {\displaystyle W}$ is a subspace of $V {\displaystyle V}$ then thequotient space $V/W$ is a vector space in its own right, and so has a dual. By thefirst isomorphism theorem, a functional $f:V\to F$ factors through $V/W$ if and only if $W {\displaystyle W}$ is in thekernel of $f {\displaystyle f}$ . There is thus an isomorphism

(V/W)^{*}\cong W^{0}.

As a particular consequence, if $V {\displaystyle V}$ is adirect sum of two subspaces $A {\displaystyle A}$ and $B {\displaystyle B}$ , then $V^{*}$ is a direct sum of $A^{0}$ and $B^{0}$ .

Dimensional analysis

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The dual space is analogous to a "negative"-dimensional space. Most simply, since a vector $v\in V$ can be paired with a covector $\varphi \in V^{*}$ by the natural pairing $\langle x,\varphi \rangle :=\varphi (x)\in F$ to obtain a scalar, a covector can "cancel" the dimension of a vector, similar toreducing a fraction. Thus while the direct sum $V\oplus V^{*}$ is a⁠ $2n$ ⁠-dimensional space (if⁠ $V {\displaystyle V}$ ⁠ is⁠ $n {\displaystyle n}$ ⁠-dimensional),⁠ $V^{*}$ ⁠ behaves as an⁠ $(-n)$ ⁠-dimensional space, in the sense that its dimensions can be canceled against the dimensions of⁠ $V {\displaystyle V}$ ⁠. This is formalized bytensor contraction.

This arises in physics viadimensional analysis, where the dual space has inverse units.^[13] Under the natural pairing, these units cancel, and the resulting scalar value isdimensionless, as expected. For example, in (continuous)Fourier analysis, or more broadlytime–frequency analysis:^{[nb 4]} given a one-dimensional vector space with aunit of time⁠ $t {\displaystyle t}$ ⁠, the dual space has units offrequency: occurrencesper unit of time (units of⁠ $1/t$ ⁠). For example, if time is measured inseconds, the corresponding dual unit is theinverse second: over the course of 3 seconds, an event that occurs 2 times per second occurs a total of 6 times, corresponding to $3s\cdot 2s^{-1}=6$ . Similarly, if the primal space measures length, the dual space measuresinverse length.

Continuous dual space

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When dealing withtopological vector spaces, thecontinuous linear functionals from the space into the base field $\mathbb {F} =\mathbb {C}$ (or $\mathbb {R}$ ) are particularly important.This gives rise to the notion of the "continuous dual space" or "topological dual" which is a linear subspace of the algebraic dual space $V^{*}$ , denoted by $V^{'} {\displaystyle V'}$ .For anyfinite-dimensional normed vector space or topological vector space, such asEuclideann-space, the continuous dual and the algebraic dual coincide.This is however false for any infinite-dimensional normed space, as shown by the example ofdiscontinuous linear maps.Nevertheless, in the theory oftopological vector spaces the terms "continuous dual space" and "topological dual space" are often replaced by "dual space".

For atopological vector space $V {\displaystyle V}$ itscontinuous dual space,^[14] ortopological dual space,^[15] or justdual space^[14]^[15]^[16]^[17] (in the sense of the theory of topological vector spaces) $V^{'} {\displaystyle V'}$ is defined as the space of all continuous linear functionals $\varphi :V\to {\mathbb {F} }$ .

Important examples for continuous dual spaces are the space of compactly supportedtest functions ${\mathcal {D}}$ and its dual ${\mathcal {D}}',$ the space of arbitrarydistributions (generalized functions); the space of arbitrary test functions ${\mathcal {E}}$ and its dual ${\mathcal {E}}',$ the space of compactly supported distributions; and the space of rapidly decreasing test functions ${\mathcal {S}},$ theSchwartz space, and its dual ${\mathcal {S}}',$ the space oftempered distributions (slowly growing distributions) in the theory ofgeneralized functions.

Properties

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IfX is aHausdorff topological vector space (TVS), then the continuous dual space ofX is identical to the continuous dual space of thecompletion ofX.^[1]

Topologies on the dual

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Main articles:Polar topology andDual system

There is a standard construction for introducing a topology on the continuous dual $V^{'} {\displaystyle V'}$ of a topological vector space $V {\displaystyle V}$ . Fix a collection ${\mathcal {A}}$ ofbounded subsets of $V {\displaystyle V}$ .This gives the topology on $V {\displaystyle V}$ ofuniform convergence on sets from ${\mathcal {A}},$ or what is the same thing, the topology generated byseminorms of the form

\|\varphi \|_{A}=\sup _{x\in A}|\varphi (x)|,

where $\varphi$ is a continuous linear functional on $V {\displaystyle V}$ , and $A {\displaystyle A}$ runs over the class ${\mathcal {A}}.$

This means that a net of functionals $\varphi _{i}$ tends to a functional $\varphi$ in $V^{'} {\displaystyle V'}$ if and only if

{\text{ for all }}A\in {\mathcal {A}}\qquad \|\varphi _{i}-\varphi \|_{A}=\sup _{x\in A}|\varphi _{i}(x)-\varphi (x)|{\underset {i\to \infty }{\longrightarrow }}0.

Usually (but not necessarily) the class ${\mathcal {A}}$ is supposed to satisfy the following conditions:

Each point $x {\displaystyle x}$ of $V {\displaystyle V}$ belongs to some set $A\in {\mathcal {A}}$ :
${\text{ for all }}x\in V\quad {\text{ there exists some }}A\in {\mathcal {A}}\quad {\text{ such that }}x\in A.$
Each two sets $A\in {\mathcal {A}}$ and $B\in {\mathcal {A}}$ are contained in some set $C\in {\mathcal {A}}$ :
${\text{ for all }}A,B\in {\mathcal {A}}\quad {\text{ there exists some }}C\in {\mathcal {A}}\quad {\text{ such that }}A\cup B\subseteq C.$
${\mathcal {A}}$ is closed under the operation of multiplication by scalars:
${\text{ for all }}A\in {\mathcal {A}}\quad {\text{ and all }}\lambda \in {\mathbb {F} }\quad {\text{ such that }}\lambda \cdot A\in {\mathcal {A}}.$

If these requirements are fulfilled then the corresponding topology on $V^{'} {\displaystyle V'}$ is Hausdorff and the sets

U_{A}~=~\left\{\varphi \in V'~:~\quad \|\varphi \|_{A}<1\right\},\qquad {\text{ for }}A\in {\mathcal {A}}

form its local base.

Here are the three most important special cases.

Thestrong topology on $V^{'} {\displaystyle V'}$ is the topology of uniform convergence onbounded subsets in $V {\displaystyle V}$ (so here ${\mathcal {A}}$ can be chosen as the class of all bounded subsets in $V {\displaystyle V}$ ).

If $V {\displaystyle V}$ is anormed vector space (for example, aBanach space or aHilbert space) then the strong topology on $V^{'} {\displaystyle V'}$ is normed (in fact a Banach space if the field of scalars is complete), with the norm

\|\varphi \|=\sup _{\|x\|\leq 1}|\varphi (x)|.

Thestereotype topology on $V^{'} {\displaystyle V'}$ is the topology of uniform convergence ontotally bounded sets in $V {\displaystyle V}$ (so here ${\mathcal {A}}$ can be chosen as the class of all totally bounded subsets in $V {\displaystyle V}$ ).
Theweak topology on $V^{'} {\displaystyle V'}$ is the topology of uniform convergence on finite subsets in $V {\displaystyle V}$ (so here ${\mathcal {A}}$ can be chosen as the class of all finite subsets in $V {\displaystyle V}$ ).

Each of these three choices of topology on $V^{'} {\displaystyle V'}$ leads to a variant ofreflexivity property for topological vector spaces:

If $V^{'} {\displaystyle V'}$ is endowed with thestrong topology, then the corresponding notion of reflexivity is the standard one: the spaces reflexive in this sense are just calledreflexive.^[18]
If $V^{'} {\displaystyle V'}$ is endowed with the stereotype dual topology, then the corresponding reflexivity is presented in the theory ofstereotype spaces: the spaces reflexive in this sense are calledstereotype.
If $V^{'} {\displaystyle V'}$ is endowed with theweak topology, then the corresponding reflexivity is presented in the theory ofdual pairs:^[19] the spaces reflexive in this sense are arbitrary (Hausdorff) locally convex spaces with the weak topology.^[20]

Examples

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Let 1 <p < ∞ be a real number and consider the Banach spaceℓ^p of allsequencesa = (a_n) for which

\|\mathbf {a} \|_{p}=\left(\sum _{n=0}^{\infty }|a_{n}|^{p}\right)^{\frac {1}{p}}<\infty .

Define the numberq by1/p + 1/q = 1. Then the continuous dual ofℓ^p is naturally identified withℓ^q: given an element $\varphi \in (\ell ^{p})'$ , the corresponding element ofℓ^q is the sequence $(\varphi (\mathbf {e} _{n}))$ where $\mathbf {e} _{n}$ denotes the sequence whosen-th term is 1 and all others are zero. Conversely, given an elementa = (a_n) ∈ℓ^q, the corresponding continuous linear functional $\varphi$ onℓ^p is defined by

\varphi (\mathbf {b} )=\sum _{n}a_{n}b_{n}

for allb = (b_n) ∈ℓ^p (seeHölder's inequality).

In a similar manner, the continuous dual ofℓ¹ is naturally identified withℓ^∞ (the space of bounded sequences).Furthermore, the continuous duals of the Banach spacesc (consisting of allconvergent sequences, with thesupremum norm) andc₀ (the sequences converging to zero) are both naturally identified withℓ¹.

By theRiesz representation theorem, the continuous dual of a Hilbert space is again a Hilbert space which isanti-isomorphic to the original space.This gives rise to thebra–ket notation used by physicists in the mathematical formulation ofquantum mechanics.

By theRiesz–Markov–Kakutani representation theorem, the continuous dual of certain spaces of continuous functions can be described using measures.

Transpose of a continuous linear map

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IfT :V → W is a continuous linear map between two topological vector spaces, then the (continuous) transposeT′ :W′ → V′ is defined by the same formula as before:

T'(\varphi )=\varphi \circ T,\quad \varphi \in W'.

The resulting functionalT′(φ) is inV′. The assignmentT → T′ produces a linear map between the space of continuous linear maps fromV toW and the space of linear maps fromW′ toV′.WhenT andU are composable continuous linear maps, then

(U\circ T)'=T'\circ U'.

WhenV andW are normed spaces, the norm of the transpose inL(W′,V′) is equal to that ofT inL(V,W).Several properties of transposition depend upon theHahn–Banach theorem.For example, the bounded linear mapT has dense rangeif and only if the transposeT′ is injective.

WhenT is acompact linear map between two Banach spacesV andW, then the transposeT′ is compact.This can be proved using theArzelà–Ascoli theorem.

WhenV is a Hilbert space, there is an antilinear isomorphismi_V fromV onto its continuous dualV′.For every bounded linear mapT onV, the transpose and theadjoint operators are linked by

i_{V}\circ T^{*}=T'\circ i_{V}.

WhenT is a continuous linear map between two topological vector spacesV andW, then the transposeT′ is continuous whenW′ andV′ are equipped with "compatible" topologies: for example, when forX =V andX =W, both dualsX′ have thestrong topologyβ(X′,X) of uniform convergence on bounded sets ofX, or both have the weak-∗ topologyσ(X′,X) of pointwise convergence on X.The transposeT′ is continuous fromβ(W′,W) toβ(V′,V), or fromσ(W′,W) toσ(V′,V).

Annihilators

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Assume thatW is a closed linear subspace of a normed space V, and consider the annihilator ofW inV′,

W^{\perp }=\{\varphi \in V':W\subseteq \ker \varphi \}.

Then, the dual of the quotientV / W can be identified withW^⊥, and the dual ofW can be identified with the quotientV′ / W^⊥.^[21]Indeed, letP denote the canonicalsurjection fromV onto the quotientV / W ; then, the transposeP′ is an isometric isomorphism from(V / W )′ intoV′, with range equal toW^⊥.Ifj denotes the injection map fromW intoV, then the kernel of the transposej′ is the annihilator ofW:

\ker(j')=W^{\perp }

and it follows from theHahn–Banach theorem thatj′ induces an isometric isomorphismV′ / W^⊥ →W′.

Further properties

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If the dual of a normed spaceV isseparable, then so is the spaceV itself.The converse is not true: for example, the spaceℓ¹ is separable, but its dualℓ^∞ is not.

Double dual

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{\displaystyle [\Psi (x_{1})+'\Psi (x_{2})](\varphi )=\varphi (x_{1}+x_{2})=\varphi (x)} — This is anatural transformation of vector addition from a vector space to its double dual.⟨x₁,x₂⟩ denotes theordered pair of two vectors. The addition + sendsx₁ andx₂ tox₁ +x₂. The addition +′ induced by the transformation can be defined as $[\Psi (x_{1})+'\Psi (x_{2})](\varphi )=\varphi (x_{1}+x_{2})=\varphi (x)$ for any $\varphi$ in the dual space.

In analogy with the case of the algebraic double dual, there is always a naturally defined continuous linear operatorΨ :V →V′′ from a normed spaceV into its continuous double dualV′′, defined by

\Psi (x)(\varphi )=\varphi (x),\quad x\in V,\ \varphi \in V'.

As a consequence of theHahn–Banach theorem, this map is in fact anisometry, meaning‖ Ψ(x) ‖ = ‖x ‖ for allx ∈V.Normed spaces for which the map Ψ is abijection are calledreflexive.

WhenV is atopological vector space then Ψ(x) can still be defined by the same formula, for everyx ∈V, however several difficulties arise.First, whenV is notlocally convex, the continuous dual may be equal to { 0 } and the map Ψ trivial.However, ifV isHausdorff and locally convex, the map Ψ is injective fromV to the algebraic dualV′^∗ of the continuous dual, again as a consequence of the Hahn–Banach theorem.^{[nb 5]}

Second, even in the locally convex setting, several natural vector space topologies can be defined on the continuous dualV′, so that the continuous double dualV′′ is not uniquely defined as a set. Saying that Ψ maps fromV toV′′, or in other words, that Ψ(x) is continuous onV′ for everyx ∈V, is a reasonable minimal requirement on the topology ofV′, namely that the evaluation mappings

\varphi \in V'\mapsto \varphi (x),\quad x\in V,

be continuous for the chosen topology onV′. Further, there is still a choice of a topology onV′′, and continuity of Ψ depends upon this choice.As a consequence, definingreflexivity in this framework is more involved than in the normed case.

Notes

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^For $V^{\lor }$ used in this way, seeAn Introduction to Manifolds (Tu 2011, p. 19).This notation is sometimes used when $(\cdot )^{*}$ is reserved for some other meaning.For instance, in the above text, $F^{*}$ is frequently used to denote the codifferential of $F {\displaystyle F}$ , so that $F^{*}\omega$ represents the pullback of the form $\omega$ .Halmos (1974, p. 20) uses $V^{'} {\displaystyle V'}$ to denote the algebraic dual of $V {\displaystyle V}$ . However, other authors use $V^{'} {\displaystyle V'}$ for the continuous dual, while reserving $V^{*}$ for the algebraic dual (Trèves 2006, p. 35).
^In many areas, such asquantum mechanics, $\langle \cdot ,\cdot \rangle$ is reserved for asesquilinear form defined on $V\times V$ .
^^a ^b ^c ^dSeveral assertions in this article require theaxiom of choice for their justification. The axiom of choice is needed to show that an arbitrary vector space has a basis: in particular it is needed to show that $\mathbb {R} ^{\mathbb {N} }$ has a basis.It is also needed to show that the dual of an infinite-dimensional vector space $V {\displaystyle V}$ is nonzero, and hence that the natural map from $V {\displaystyle V}$ to its double dual is injective.
^To be precise, continuous Fourier analysis studies the space offunctionals with domain a vector space and the space of functionals on the dual vector space.
^IfV is locally convex but not Hausdorff, thekernel of Ψ is the smallest closed subspace containing {0}.

References

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^^a ^bNarici & Beckenstein 2011, pp. 225–273.
^Katznelson & Katznelson (2008) p. 37, §2.1.3
^^a ^bTu (2011) p. 19, §3.1
^Axler (2015) p. 101, §3.94
^Halmos (1974) p. 20, §13
^Halmos (1974) p. 21, §14
^Misner, Thorne & Wheeler 1973
^Misner, Thorne & Wheeler 1973, §2.5
^Nicolas Bourbaki (1974). Hermann (ed.).Elements of mathematics: Algebra I, Chapters 1 - 3. Addison-Wesley Publishing Company. p. 400.ISBN 0201006391.
^Mac Lane & Birkhoff 1999, §VI.4
^Halmos (1974) pp. 25, 28
^Halmos (1974) §44
^Tao, Terence (2012-12-29)."A mathematical formalisation of dimensional analysis".Similarly, one can define $V^{T^{-1}}$ as the dual space to $V^{T}$ ...
^^a ^bRobertson & Robertson 1964, II.2
^^a ^bSchaefer 1966, II.4
^Rudin 1973, 3.1
^Bourbaki 2003, II.42
^Schaefer 1966, IV.5.5
^Schaefer 1966, IV.1
^Schaefer 1966, IV.1.2
^Rudin 1991, chapter 4

Bibliography

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Axler, Sheldon Jay (2015).Linear Algebra Done Right (3rd ed.).Springer.ISBN 978-3-319-11079-0.
Bourbaki, Nicolas (1989).Elements of mathematics, Algebra I. Springer-Verlag.ISBN 3-540-64243-9.
Bourbaki, Nicolas (2003).Elements of mathematics, Topological vector spaces. Springer-Verlag.
Halmos, Paul Richard (1974) [1958].Finite-Dimensional Vector Spaces (2nd ed.).Springer.ISBN 0-387-90093-4.
Katznelson, Yitzhak; Katznelson, Yonatan R. (2008).A (Terse) Introduction to Linear Algebra.American Mathematical Society.ISBN 978-0-8218-4419-9.
Lang, Serge (2002),Algebra,Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag,ISBN 978-0-387-95385-4,MR 1878556,Zbl 0984.00001
Tu, Loring W. (2011).An Introduction to Manifolds (2nd ed.).Springer.ISBN 978-1-4419-7400-6.
Mac Lane, Saunders;Birkhoff, Garrett (1999).Algebra (3rd ed.). AMS Chelsea Publishing.ISBN 0-8218-1646-2..
Misner, Charles W.;Thorne, Kip S.;Wheeler, John A. (1973).Gravitation. W. H. Freeman.ISBN 0-7167-0344-0.
Narici, Lawrence; Beckenstein, Edward (2011).Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.ISBN 978-1584888666.OCLC 144216834.
Rudin, Walter (1973).Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 25 (First ed.). New York, NY:McGraw-Hill Science/Engineering/Math.ISBN 9780070542259.
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.
Robertson, A.P.; Robertson, W. (1964).Topological vector spaces. Cambridge University Press.
Schaefer, Helmut H. (1966).Topological vector spaces. New York: The Macmillan Company.
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.

External links

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Weisstein, Eric W."Dual Vector Space".MathWorld.

v t e Linear algebra
Outline Glossary Template:Matrix classes
Linear equations	Linear equation System of linear equations Determinant Minor Cauchy–Binet formula Cramer's rule Gaussian elimination Gauss–Jordan elimination Overcompleteness Strassen algorithm
Matrices	Matrix Matrix addition Matrix multiplication Basis transformation matrix Characteristic polynomial Spectrum Trace Eigenvalue, eigenvector and eigenspace Cayley–Hamilton theorem Jordan normal form Weyr canonical form Rank Inverse,Pseudoinverse Adjugate,Transpose Dot product Symmetric matrix,Skew-symmetric matrix Orthogonal matrix,Unitary matrix Hermitian matrix,Antihermitian matrix Positive-(semi)definite Pfaffian Projection Spectral theorem Perron–Frobenius theorem Diagonal matrix,Triangular matrix,Tridiagonal matrix Block matrix Sparse matrix Hessenberg matrix,Hessian matrix Vandermonde matrix Stochastic matrix,Toeplitz matrix,Circulant matrix,Hankel matrix (0,1)-matrix List of matrices
Matrix decompositions	Cholesky decomposition LU decomposition QR decomposition Polar decomposition Spectral theorem Singular value decomposition Higher-order singular value decomposition Schur decomposition Schur complement Haynsworth inertia additivity formula Reducing subspace
Relations and computations	Matrix equivalence Matrix congruence Matrix similarity Matrix consimilarity Row equivalence Elementary row operations Householder transformation Least squares Linear least squares Gram–Schmidt process Woodbury matrix identity
Vector spaces	Vector space Linear combination Linear span Linear independence Basis,Hamel basis Change of basis Dimension theorem for vector spaces Hamel dimension Examples of vector spaces Linear map Shear mapping Squeeze mapping Linear subspace Row and column spaces,Null space Rank–nullity theorem Nullity theorem Cyclic subspace Dual space,Linear functional Category of vector spaces
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v t e Duality and spaces oflinear maps
Basic concepts	Dual space Dual system Dual topology Duality Operator topologies Polar set Polar topology Topologies on spaces of linear maps
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Subsets	Saturated family Total set
Other concepts	Biorthogonal system