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| In [[mathematics]], any [[vector space]], ''V'', has a corresponding '''dual vector space''' (or just '''dual space''' for short) consisting of all [[linear functional]]s on ''V''. Dual vector spaces for finite-dimensional vector spaces can be used for studying [[tensors]]. When applied to vector spaces of functions (which are typically infinite-dimensional), dual spaces are employed for defining and studying concepts like [[Measure (mathematics)|measures]], [[Distribution (mathematics)|distributions]], and [[Hilbert space]]s. Consequently, the dual space is an important concept in the study of [[functional analysis]].
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| There are two types of dual spaces: the ''algebraic dual space'', and the ''continuous dual space''. The algebraic dual space is defined for all vector spaces. When defined for a [[topological vector space]] there is a subspace of this dual space, corresponding to continuous linear functionals, which constitutes a continuous dual space.
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| == Algebraic dual space ==
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| Given any [[vector space]] ''V'' over a [[field (mathematics)|field]] ''F'', the '''dual space''' ''V*'' is defined as the set of all [[linear map]]s {{nowrap|''φ'': ''V'' → ''F''}} ([[linear functional]]s). The dual space ''V*'' itself becomes a vector space over ''F'' when equipped with the following addition and scalar multiplication:
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| : <math>\begin{align}
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| & (\varphi + \psi)(x) = \varphi(x) + \psi(x) \\
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| & (a \varphi)(x) = a \left(\varphi(x)\right)
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| \end{align}</math>
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| for all ''φ'' and ''ψ'' ∈ ''V*'', ''x'' ∈ ''V'', and ''a'' ∈ ''F''. Elements of the algebraic dual space ''V*'' are sometimes called '''covectors''' or '''[[one-form]]s'''.
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| The pairing of a functional ''φ'' in the dual space ''V*'' and an element ''x'' of ''V'' is sometimes denoted by a bracket: {{nowrap|''φ''(''x'') {{=}} [''φ'',''x'']}} <ref>{{harvtxt|Halmos|1974}}</ref>
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| or {{nowrap|''φ''(''x'') {{=}} ⟨''φ'',''x''⟩}}.<ref>{{harvtxt|Misner|Thorne|Wheeler|1973}}</ref> The pairing defines a nondegenerate [[bilinear mapping]]<ref>In many areas, such as [[quantum mechanics]], ⟨·,·⟩ is reserved for a [[sesquilinear form]] defined on {{nowrap|''V × V''}}.</ref> {{nowrap|[·,·] : ''V* × V'' → ''F''}}.
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| === Finite-dimensional case ===
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| If ''V'' is [[finite-dimensional]], then ''V*'' has the same dimension as ''V''. Given a [[basis of a vector space|basis]] {{nowrap|{'''e'''<sub>1</sub>, ..., '''e'''<sub>''n''</sub>}}} in ''V'', it is possible to construct a specific basis in ''V*'', called the '''[[dual basis]]'''. This dual basis is a set {{nowrap|{'''e'''<sup>1</sup>, ..., '''e'''<sup>''n''</sup>}}} of linear functionals on ''V'', defined by the relation
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| : <math>
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| \mathbf{e}^i(c_1 \mathbf{e}_1+\cdots+c_n\mathbf{e}_n) = c_i, \quad i=1,\ldots,n
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| </math>
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| for any choice of coefficients {{nowrap|''c<sub>i</sub>'' ∈ ''F''}}. In particular, letting in turn each one of those coefficients be equal to one and the other coefficients zero, gives the system of equations
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| : <math>
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| \mathbf{e}^i(\mathbf{e}_j) = \delta_{ij}
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| </math>
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| where ''δ<sub>ij</sub>'' is the [[Kronecker delta]] symbol. For example if ''V'' is '''R'''<sup>2</sup>, and its basis chosen to be {{nowrap|{'''e'''<sub>1</sub> {{=}} (1, 0), '''e'''<sub>2</sub> {{=}} (0, 1)}}}, then '''e'''<sup>1</sup> and '''e'''<sup>2</sup> are [[one-form]]s (functions which map a vector to a scalar) such that {{nowrap|'''e'''<sup>1</sup>('''e'''<sub>1</sub>) {{=}} 1}}, {{nowrap|'''e'''<sup>1</sup>('''e'''<sub>2</sub>) {{=}} 0}}, {{nowrap|'''e'''<sup>2</sup>('''e'''<sub>1</sub>) {{=}} 0}}, and {{nowrap|'''e'''<sup>2</sup>('''e'''<sub>2</sub>) {{=}} 1}}. (Note: The superscript here is the index, not an exponent).
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| In particular, if we interpret '''R'''<sup>''n''</sup> as the space of columns of ''n'' [[real number]]s, its dual space is typically written as the space of ''rows'' of ''n'' real numbers. Such a row acts on '''R'''<sup>''n''</sup> as a linear functional by ordinary [[matrix multiplication]].
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| If ''V'' consists of the space of geometrical [[Vector (geometric)|vector]]s (arrows) in the plane, then the level curves of an element of ''V*'' form a family of parallel lines in ''V''. So an element of ''V*'' can be intuitively thought of as a particular family of parallel lines covering the plane. To compute the value of a functional on a given vector, one needs only to determine which of the lines the vector lies on. Or, informally, one "counts" how many lines the vector crosses. More generally, if ''V'' is a vector space of any dimension, then the level sets of a linear functional in ''V*'' are parallel hyperplanes in ''V'', and the action of a linear functional on a vector can be visualized in terms of these hyperplanes.<ref>{{harvtxt|Misner|Thorne|Wheeler|1973|loc=§2.5}}</ref>
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| === Infinite-dimensional case ===
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| If ''V'' is not finite-dimensional but has a [[basis (linear algebra)|basis]]<ref name="choice">Several assertions in this article require the [[axiom 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 '''R<sup>N</sup>''' has a basis. It is also needed to show that the dual of an infinite-dimensional vector space ''V'' is nonzero, and hence that the natural map from ''V'' to its double dual is injective.</ref> '''e'''<sub>''α''</sub> indexed by an infinite set ''A'', then the same construction as in the finite-dimensional case yields [[linearly independent]] elements '''e'''<sup>''α''</sup> ({{nowrap|''α'' ∈ ''A''}}) of the dual space, but they will not form a basis.
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| Consider, for instance, the space '''R'''<sup>∞</sup>, whose elements are those [[sequence]]s of real numbers which have only finitely many non-zero entries, which has a basis indexed by the natural numbers '''N''': for {{nowrap|''i'' ∈ '''N'''}}, '''e'''<sub>''i''</sub> is the sequence which is zero apart from the ''i''th term, which is one. The dual space of '''R'''<sup>∞</sup> is '''R<sup>N</sup>''', the space of ''all'' sequences of real numbers: such a sequence (''a<sub>n</sub>'') is applied to an element (''x<sub>n</sub>'') of '''R'''<sup>∞</sup> to give the number ∑''a<sub>n</sub>x<sub>n</sub>'', which is a finite sum because there are only finitely many nonzero ''x<sub>n</sub>''. The [[dimension (vector space)|dimension]] of '''R'''<sup>∞</sup> is countably infinite, whereas '''R<sup>N</sup>''' does not have a countable basis.
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| This observation generalizes to any<ref name="choice"/> infinite-dimensional vector space ''V'' over any field ''F'': a choice of basis {{nowrap|{'''e'''<sub>''α''</sub> : ''α'' ∈ ''A''}}} identifies ''V'' with the space (''F<sup>A</sup>'')<sub>0</sub> of functions {{nowrap|''f'' : ''A → F''}} such that {{nowrap|''f<sub>α</sub>'' {{=}} ''f''(''α'')}} is nonzero for only finitely many {{nowrap|''α'' ∈ ''A''}}, where such a function ''f'' is identified with the vector
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| : <math>
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| \sum_{\alpha\in A} f_\alpha\mathbf{e}_\alpha
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| </math>
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| in ''V'' (the sum is finite by the assumption on ''f'', and any {{nowrap|''v'' ∈ ''V''}} may be written in this way by the definition of the basis).
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| The dual space of ''V'' may then be identified with the space ''F<sup>A</sup>'' of ''all'' functions from ''A'' to ''F'': a linear functional ''T'' on ''V'' is uniquely determined by the values {{nowrap|''θ<sub>α</sub>'' {{=}} ''T''('''e'''<sub>''α''</sub>)}} it takes on the basis of ''V'', and any function {{nowrap|''θ'' : ''A → F''}} (with {{nowrap|''θ''(''α'') {{=}} ''θ<sub>α</sub>''}}) defines a linear functional ''T'' on ''V'' by
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| : <math>
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| T\biggl(\sum_{\alpha\in A} f_\alpha \mathbf{e}_\alpha\biggr) = \sum_{\alpha \in A} f_\alpha T(e_\alpha) = \sum_{\alpha\in A} f_\alpha \theta_\alpha.
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| </math>
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| Again the sum is finite because ''f<sub>α</sub>'' is nonzero for only finitely many ''α''.
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| Note that (''F<sup>A</sup>'')<sub>0</sub> may be identified (essentially by definition) with the [[Direct sum of modules|direct sum]]
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| of infinitely many copies of ''F'' (viewed as a 1-dimensional vector space over itself) indexed by ''A'', i.e., there are linear isomorphisms
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| : <math>
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| V\cong (F^A)_0\cong\bigoplus_{\alpha\in A} {F}.
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| </math>
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| On the other hand ''F<sup>A</sup>'' is (again by definition), the [[direct product]] of infinitely many copies of ''F'' indexed by ''A'', and so the identification
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| : <math>
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| V^* \cong
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| \biggl(\bigoplus_{\alpha\in A}F\biggr)^* \cong
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| \prod_{\alpha\in A}F^* \cong
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| \prod_{\alpha\in A}F \cong
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| F^A
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| </math>
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| is a special case of a [[Direct sum of modules#Properties|general result]] relating direct sums (of modules) to direct products.
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| Thus if the basis is infinite, then the algebraic dual space is ''always'' of larger dimension than the original vector space. This is in marked contrast to the case of the continuous dual space, discussed below, which may be [[isomorphic]] to the original vector space even if the latter is infinite-dimensional.
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| === Bilinear products and dual spaces ===
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| If ''V'' is finite-dimensional, then ''V'' is isomorphic to ''V*''. But there is in general no [[natural isomorphism]] between these two spaces.<ref>{{harvtxt|MacLane|Birkhoff|1999|loc=§VI.4}}</ref> Any [[bilinear form]] ⟨•,•⟩ on ''V'' gives a mapping of ''V'' into its dual space via
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| :<math>v\mapsto \langle v, \cdot\rangle</math> | |
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| where the right hand side is defined as the functional on ''V'' taking each {{nowrap|''w'' ∈ ''V''}} to ⟨''v'',''w''⟩. In other words, the bilinear form determines a linear mapping
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| :<math>\Phi_{\langle\cdot,\cdot\rangle} : V\to V^*</math>
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| defined by
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| :<math>[\Phi_{\langle\cdot,\cdot\rangle}(v),w] = \langle v, w\rangle.</math>
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| If the bilinear form is [[nondegenerate form|nondegenerate]], then this is an isomorphism onto a subspace of ''V*''. If ''V'' is finite-dimensional, then this is an isomorphism onto all of ''V*''. Conversely, any isomorphism Φ from ''V'' to a subspace of ''V*'' (resp., all of ''V*'') defines a unique nondegenerate bilinear form ⟨•,•⟩<sub>Φ</sub> on ''V'' by
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| :<math> \langle v,w \rangle_\Phi = (\Phi (v))(w) = [\Phi (v),w].\,</math>
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| Thus there is a one-to-one correspondence between isomorphisms of ''V'' to subspaces of (resp., all of) ''V*'' and nondegenerate bilinear forms on ''V''.
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| If the vector space ''V'' is over the [[complex numbers|complex]] field, then sometimes it is more natural to consider [[sesquilinear form]]s instead of bilinear forms. In that case, a given sesquilinear form ⟨•,•⟩ determines an isomorphism of ''V'' with the [[Complex conjugate vector space|complex conjugate]] of the dual space
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| : <math>
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| \Phi_{\langle\cdot,\cdot\rangle} : V\to \overline{V}^*.
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| </math>
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| The conjugate space ''<span style="text-decoration:overline">V</span>*'' can be identified with the set of all additive complex-valued functionals {{nowrap|''f'' : ''V'' → '''C'''}} such that
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| : <math>
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| f(\alpha v) = \overline{\alpha}f(v).
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| </math>
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| === Injection into the double-dual ===
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| There is a [[natural transformation|natural]] [[linear map|homomorphism]] Ψ from ''V'' into the double dual ''V**'', defined by {{nowrap|(Ψ(''v''))(''φ'') {{=}} ''φ''(''v'')}} for all {{nowrap|''v'' ∈ ''V''}}, {{nowrap|''φ'' ∈ ''V*''}}. This map Ψ is always [[injective]];<ref name="choice"/> it is an [[isomorphism]] if and only if ''V'' is finite-dimensional. Indeed, the isomorphism of a finite-dimensional vector space with its double dual is an archetypal example of a [[natural isomorphism]]. Note that infinite-dimensional Hilbert spaces are not a counterexample to this, as they are isomorphic to their continuous duals, not to their algebraic duals.
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| === Transpose of a linear map === <!-- this section is linked to in [[matrix (mathematics)]] and [[transpose]] -->
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| If {{nowrap|''f'' : ''V → W''}} is a [[linear map]], then the ''[[Transpose#Transpose of linear maps|transpose]]'' (or ''dual'') {{nowrap|''f*'' : ''W* → V*''}} is defined by
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| : <math>
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| f^*(\varphi) = \varphi \circ f \,
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| </math>
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| for every {{nowrap|''φ'' ∈ ''W*''}}. The resulting functional ''f*''(''φ'') in ''V*'' is called the ''[[pullback (differential geometry)|pullback]]'' of ''φ'' along ''f''.
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| The following identity holds for all {{nowrap|''φ'' ∈ ''W*''}} and {{nowrap|''v'' ∈ ''V''}}:
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| : <math>
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| [f^*(\varphi),\, v] = [\varphi,\, f(v)],
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| </math>
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| where the bracket [•,•] on the left is the duality pairing of ''V'' with its dual space, and that on the right is the duality pairing of ''W'' with its dual. This identity characterizes the transpose,<ref>{{harvtxt|Halmos|1974|loc=§44}}</ref> and is formally similar to the definition of the [[adjoint of an operator|adjoint]].
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| The assignment {{nowrap|''f'' ↦ ''f*''}} produces an [[injective]] linear map between the space of linear operators from ''V'' to ''W'' and the space of linear operators from ''W*'' to ''V*''; this homomorphism is an [[isomorphism]] if and only if ''W'' is finite-dimensional. If {{nowrap|''V'' {{=}} ''W''}} then the space of linear maps is actually an [[algebra over a field|algebra]] under [[composition of maps]], and the assignment is then an [[antihomomorphism]] of algebras, meaning that {{nowrap|(''fg'')* {{=}} ''g*f*''}}. 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 ''F'' to itself. Note that one can identify (''f*'')* with ''f'' using the natural injection into the double dual.
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| If the linear map ''f'' is represented by the [[matrix (mathematics)|matrix]] ''A'' with respect to two bases of ''V'' and ''W'', then ''f*'' is represented by the [[transpose]] matrix ''A''<sup>T</sup> with respect to the dual bases of ''W*'' and ''V*'', hence the name. Alternatively, as ''f'' is represented by ''A'' 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 on '''R'''<sup>''n''</sup>, which identifies the space of column vectors with the dual space of row vectors.
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| === Quotient spaces and annihilators ===
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| Let ''S'' be a subset of ''V''. The '''[[annihilator (ring theory)|annihilator]]''' of ''S'' in ''V*'', denoted here ''S<sup>o</sup>'', is the collection of linear functionals {{nowrap|''f'' ∈ ''V*''}} such that {{nowrap|[''f'', ''s''] {{=}} 0}} for all {{nowrap|''s'' ∈ ''S''}}. That is, ''S<sup>o</sup>'' consists of all linear functionals {{nowrap|''f'' : ''V → F''}} such that the restriction to ''S'' vanishes: {{nowrap|''f''{{!}}<sub>''S''</sub> {{=}} 0}}.
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| The annihilator of a subset is itself a vector space. In particular, {{nowrap|∅<sup>''o''</sup> {{=}} ''V*''}} is all of ''V*'' ([[vacuously true|vacuously]]), whereas {{nowrap|''V<sup>o</sup>'' {{=}} 0}} is the zero subspace. Furthermore, the assignment of an annihilator to a subset of ''V'' reverses inclusions, so that if {{nowrap|''S'' ⊂ ''T'' ⊂ ''V''}}, then
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| : <math>
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| 0 \subset T^o \subset S^o \subset V^*.
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| </math>
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| Moreover, if ''A'' and ''B'' are two subsets of ''V'', then
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| : <math>
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| (A \cap B)^o \supseteq A^o + B^o,
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| </math>
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| and equality holds provided ''V'' is finite-dimensional. If ''A<sub>i</sub>'' is any family of subsets of ''V'' indexed by ''i'' belonging to some index set ''I'', then
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| : <math>
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| \left(\bigcup_{i\in I} A_i\right)^o = \bigcap_{i\in I} A_i^o.
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| </math>
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| In particular if ''A'' and ''B'' are subspaces of ''V'', it follows that
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| : <math>
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| (A + B)^o = A^o \cap B^o.\,
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| </math>
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| If ''V'' is finite-dimensional, and ''W'' is a [[vector subspace]], then
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| : <math>
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| W^{oo} = W \,
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| </math>
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| after identifying ''W'' with its image in the second dual space under the double duality isomorphism {{nowrap|''V'' ≈ ''V**''}}. Thus, in particular, forming the annihilator is a [[Galois connection]] on the lattice of subsets of a finite-dimensional vector space.
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| If ''W'' is a subspace of ''V'' then the [[quotient space (linear algebra)|quotient space]] ''V''/''W'' is a vector space in its own right, and so has a dual. By the [[first isomorphism theorem]], a functional {{nowrap|''f'' : ''V → F''}} factors through ''V''/''W'' if and only if ''W'' is in the [[kernel (algebra)|kernel]] of ''f''. There is thus an isomorphism
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| : <math>
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| (V/W)^* \cong W^o.
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| </math>
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| As a particular consequence, if ''V'' is a [[Direct sum of modules|direct sum]] of two subspaces ''A'' and ''B'', then ''V*'' is a direct sum of ''A<sup>o</sup>'' and ''B<sup>o</sup>''.
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| == Continuous dual space ==<!-- This section is linked from [[Reflexive space]] -->
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| When dealing with [[topological vector space]]s, one is typically only interested in the [[continuous function (topology)|continuous]] linear functionals from the space into the base field <math>{\mathbb F}={\mathbb C}</math> (or <math>{\mathbb R}</math>). This gives rise to the notion of the "continuous dual space" or "topological dual" which is a linear subspace of the algebraic dual space <math>V^*</math>, denoted by <math>V'</math>. For any ''finite-dimensional'' normed vector space or topological vector space, such as [[Euclidean space|Euclidean ''n-''space]], the continuous dual and the algebraic dual coincide. This is however false for any infinite-dimensional normed space, as shown by the example of [[discontinuous linear map]]s. Nevertheless in the theory of [[topological vector space]]s the terms "continuous dual space" and "topological dual space" are rarely used, as a rule they are replaced by "dual space", since there is no serious need to consider discontinuous maps in this field.
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| For a [[topological vector space]] <math>V</math> its ''continuous dual space'',<ref name="A.P.Robertson, W.Robertson 1964 loc=II.2">{{harvtxt|A.P.Robertson, W.Robertson|1964|loc=II.2}}</ref> or ''topological dual space'',<ref name="H.Schaefer 1966 loc=II.4">{{harvtxt|H.Schaefer|1966|loc=II.4}}</ref> or just ''dual space''<ref name="A.P.Robertson, W.Robertson 1964 loc=II.2"/><ref name="H.Schaefer 1966 loc=II.4"/><ref>{{harvtxt|W.Rudin|1973|loc=3.1}}</ref><ref>{{harvtxt|Nicolas Bourbaki|2003|loc=II.42}}</ref> (in the sense of the theory of topological vector spaces) <math>V'</math> is defined as the space of all continuous linear functionals <math>\varphi:V\to{\mathbb F}</math>.
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| There is a standard construction for introducing topology on the continuous dual <math>V'</math> of a topological vector space <math>V</math>: each given class <math>{\mathcal A}</math> of [[Bounded set (topological vector space)|bounded subsets]] in <math>V</math> defines a topology on <math>V</math> of uniform convergence on sets from <math>{\mathcal A}</math>, or what is the same a topology generated by [[norm (mathematics)|seminorms]] of the form
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| : <math>
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| \|\varphi\|_A = \sup_{x\in A} |\varphi(x)|,
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| </math>
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| where <math>\varphi</math> is a continuous linear functional on <math>V</math>, and <math>A</math> runs over the class <math>{\mathcal A}</math>.
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| This means that a net of functionals <math>\varphi_i</math> tends to a functional <math>\varphi</math> in <math>V'</math> if and only if
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| : <math>
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| \forall A\in{\mathcal A}\qquad \|\varphi_i-\varphi\|_A = \sup_{x\in A} |\varphi_i(x)-\varphi(x)|\underset{i\to\infty}{\longrightarrow}0.
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| </math>
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| Usually (but not necessarily) the class <math>{\mathcal A}</math> is supposed to satisfy the following conditions:
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| * each point <math>x</math> of <math>V</math> belongs to some set <math>A\in{\mathcal A}</math>
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| : <math>
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| \forall x\in V\qquad \exists A\in {\mathcal A}\qquad x\in A,
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| </math>
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| * each two sets <math>A\in{\mathcal A}</math> and <math>B\in{\mathcal A}</math> are contained in some set <math>C\in{\mathcal A}</math>:
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| : <math>
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| \forall A,B\in {\mathcal A}\qquad \exists C\in {\mathcal A}\qquad A\cup B\subseteq C,
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| </math>
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| * <math>{\mathcal A}</math> is closed under the operation of multiplication by scalars:
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| : <math>
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| \forall A\in {\mathcal A}\qquad \forall\lambda\in{\mathbb F}\qquad \lambda\cdot A\in {\mathcal A},
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| </math>
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| If these requirements are fulfilled then the coresponding topology on <math>V'</math> is Hausdorff and the sets
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| : <math>
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| U_A=\{x\in V:\quad ||\varphi||_A<1\},\qquad A\in {\mathcal A},
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| </math>
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| form its local base.
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| Here are the three most important special cases.
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| * The [[Strong topology (polar topology)|strong topology]] on <math>V'</math> is the topology of uniform convergence on [[Bounded set (topological vector space)|bounded subsets]] in <math>V</math> (so here <math>{\mathcal A}</math> can be chosen as the class of all bounded subsets in <math>V</math>). If <math>V</math> is a [[normed vector space]] (e.g., a [[Banach space]] or a [[Hilbert space]]) then the strong topology on <math>V'</math> is normed (in fact a Banach space if the field of scalars is complete), with the norm
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| : <math>
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| \|\varphi\| = \sup_{\|x\| \le 1 } |\varphi(x)|.
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| </math>
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| * The [[stereotype space|stereotype topology]] on <math>V'</math> is the topology of uniform convergence on [[Totally bounded space|totally bounded sets]] in <math>V</math> (so here <math>{\mathcal A}</math> can be chosen as the class of all totally bounded subsets in <math>V</math>).
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| * The [[weak topology]] on <math>V'</math> is the topology of uniform convergence on finite subsets in <math>V</math> (so here <math>{\mathcal A}</math> can be chosen as the class of all finite subsets in <math>V</math>).
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| Each of these three choices of topology on <math>V'</math> leads to a variant of [[Reflexive space|reflexivity property]] for topological vector spaces.
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| === Examples ===
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| Let 1 < ''p'' < ∞ be a real number and consider the Banach space ''[[Lp_space#The_p-norm_in_countably_infinite_dimensions|ℓ<sup> p</sup>]]'' of all [[sequence]]s {{nowrap|'''a''' {{=}} (''a''<sub>''n''</sub>)}} for which
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| : <math>
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| \|\mathbf{a}\|_p = \left ( \sum_{n=0}^\infty |a_n|^p \right) ^{1/p}
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| </math>
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| is finite. Define the number ''q'' by {{nowrap|1/''p'' + 1/''q'' {{=}} 1}}. Then the continuous dual of ''ℓ<sup> p</sup>'' is naturally identified with ''ℓ<sup> q</sup>'': given an element {{nowrap|''φ'' ∈ (''ℓ<sup> p</sup>'')′}}, the corresponding element of {{nowrap|''ℓ<sup> q</sup>''}} is the sequence (''φ''('''e'''<sub>''n''</sub>)) where '''e'''<sub>''n''</sub> denotes the sequence whose ''n-''th term is 1 and all others are zero. Conversely, given an element {{nowrap|'''a''' {{=}} (''a<sub>n</sub>'') ∈ ''ℓ<sup> q</sup>''}}, the corresponding continuous linear functional ''φ'' on {{nowrap|''ℓ<sup> p</sup>''}} is defined by {{nowrap|''φ''('''b''') {{=}} ∑<sub>''n''</sub> ''a<sub>n</sub>b<sub>n</sub>''}} for all {{nowrap|'''b''' {{=}} (''b<sub>n</sub>'') ∈ ''ℓ<sup> p</sup>''}} (see [[Hölder's inequality]]).
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| In a similar manner, the continuous dual of {{nowrap|''ℓ''<sup> 1</sup>}} is naturally identified with {{nowrap|''ℓ''<sup> ∞</sup>}} (the space of bounded sequences). Furthermore, the continuous duals of the Banach spaces ''c'' (consisting of all [[limit of a sequence|convergent]] sequences, with the [[supremum norm]]) and ''c''<sub>0</sub> (the sequences converging to zero) are both naturally identified with {{nowrap|''ℓ''<sup> 1</sup>}}.
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| By the [[Riesz representation theorem]], the continuous dual of a Hilbert space is again a Hilbert space which is [[antiisomorphic|anti-isomorphic]] to the original space. This gives rise to the [[bra-ket notation]] used by physicists in the mathematical formulation of [[quantum mechanics]].
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| === Transpose of a continuous linear map ===
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| If {{nowrap|''T'' : ''V → W''}} is a continuous linear map between two topological vector spaces, then the (continuous) transpose {{nowrap|''T′'' : ''W′ → V′''}} is defined by the same formula as before:
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| : <math>
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| T'(\varphi) = \varphi \circ T, \quad \varphi \in W'. \,
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| </math>
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| The resulting functional {{nowrap|''T′''(''φ'')}} is in{{nowrap| ''V′''}}. The assignment {{nowrap|''T → T′''}} produces a linear map between the space of continuous linear maps from ''V'' to ''W'' and the space of linear maps from {{nowrap|''W′''}} to {{nowrap|''V′''}}. When ''T'' and ''U'' are composable continuous linear maps, then
| |
| : <math>
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| (U \circ T)' = T' \circ U'. \,
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| </math>
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| When ''V'' and ''W'' are normed spaces, the norm of the transpose in{{nowrap| ''L''(''W′'', ''V′'')}} is equal to that of ''T'' in{{nowrap| ''L''(''V'', ''W'')}}. Several properties of transposition depend upon the [[Hahn–Banach theorem]]. For example, the bounded linear map ''T'' has dense range if and only if the transpose {{nowrap|''T′''}} is injective.
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| When ''T'' is a [[Compact operator|compact]] linear map between two Banach spaces ''V'' and ''W'', then the transpose {{nowrap|''T′''}} is compact. This can be proved using the [[Arzelà–Ascoli theorem]].
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| When ''V'' is a Hilbert space, there is an antilinear isomorphism ''i<sub>V</sub>'' from ''V'' onto its continuous dual{{nowrap| ''V′''}}. For every bounded linear map ''T'' on ''V'', the transpose and the [[Hermitian adjoint|adjoint]] operators are linked by
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| : <math>
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| i_V \circ T^* = T' \circ i_V. \,
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| </math>
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| When ''T'' is a continuous linear map between two topological vector spaces ''V'' and ''W'', then the transpose {{nowrap|''T′''}} is continuous when {{nowrap|''W′''}} and {{nowrap|''V′''}} are equipped with"compatible" topologies: for example when, for {{nowrap|''X'' {{=}} ''V''}} and {{nowrap|''X'' {{=}} ''W''}}, both duals {{nowrap|''X′''}} have the [[Strong topology (polar topology)|strong topology]] {{nowrap|''β''(''X′'', ''X'')}} of uniform convergence on bounded sets of ''X'', or both have the weak-∗ topology {{nowrap|''σ''(''X′'', ''X'')}} of pointwise convergence on ''X''. The transpose {{nowrap|''T′''}} is continuous from {{nowrap|''β''(''W′'', ''W'')}} to {{nowrap|''β''(''V′'', ''V'')}}, or from {{nowrap|''σ''(''W′'', ''W'')}} to {{nowrap|''σ''(''V′'', ''V'')}}.
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| === Annihilators ===
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| Assume that ''W'' is a closed linear subspace of a normed space ''V'', and consider the annihilator of ''W'' in{{nowrap| ''V′''}},
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| : <math>
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| W^\perp = \{ \varphi \in V' : W \subset \ker \varphi\}. \,
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| </math>
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| Then, the dual of the quotient {{nowrap|''V'' / ''W'' }} can be identified with ''W''<sup>⊥</sup>, and the dual of ''W'' can be identified with the quotient {{nowrap|''V′'' / ''W''<sup>⊥</sup>}}.<ref>{{harvtxt|Rudin|1991|loc=chapter 4}}</ref> Indeed, let ''P'' denote the canonical [[surjection]] from ''V'' onto the quotient {{nowrap|''V'' / ''W'' }}; then, the transpose {{nowrap|''P′''}} is an isometric isomorphism from {{nowrap|(''V'' / ''W'' )′}} into{{nowrap| ''V′''}}, with range equal to ''W''<sup>⊥</sup>. If ''j'' denotes the injection map from ''W'' into ''V'', then the kernel of the transpose {{nowrap|''j′''}} is the annihilator of ''W'':
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| :<math>\ker (j') = W^\perp</math>
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| and it follows from the [[Hahn–Banach theorem]] that {{nowrap|''j′''}} induces an isometric isomorphism
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| {{nowrap|''V′'' / ''W''<sup>⊥</sup> → ''W′''}}.
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| === Further properties ===
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| If the dual of a normed space ''V'' is [[separable space|separable]], then so is the space ''V'' itself. The converse is not true: for example the space {{nowrap|''ℓ''<sup> 1</sup>}} is separable, but its dual is {{nowrap|''ℓ''<sup> ∞</sup>}} is not.
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| ===Topologies on the dual===
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| The topology of ''V'' and the topology of real or complex numbers can be used to induce on ''V′'' a [[Space_of_linear_maps#G-topologies_on_X_induced_by_the_continuous_dual|dual space topology]].
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| === Double dual ===
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| In analogy with the case of the algebraic double dual, there is always a naturally defined continuous linear operator {{nowrap|Ψ : ''V'' → ''V′′''}} from a normed space ''V'' into its continuous double dual{{nowrap| ''V′′''}}, defined by
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| : <math> | |
| \Psi(x)(\varphi) = \varphi(x), \quad x \in V, \ \varphi \in V'. \,
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| </math>
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| As a consequence of the [[Hahn–Banach theorem]], this map is in fact an [[isometry]], meaning {{nowrap|{{!!}}Ψ(''x''){{!!}} {{=}} {{!!}}''x''{{!!}}}} for all ''x'' in ''V''. Normed spaces for which the map Ψ is a [[bijection]] are called [[reflexive space|reflexive]].
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| When ''V'' is a [[topological vector space]], one can still define Ψ(''x'') by the same formula, for every {{nowrap|''x'' ∈ ''V''}}, however several difficulties arise. First, when ''V'' is not [[Locally convex topological vector space|locally convex]], the continuous dual may be equal to {0} and the map Ψ trivial. However, if ''V'' is [[Hausdorff space|Hausdorff]] and locally convex, the map Ψ is injective from ''V'' to the algebraic dual {{nowrap|''V′*''}} of the continuous dual, again as a consequence of the Hahn–Banach theorem.<ref>If ''V'' is locally convex but not Hausdorff, the [[kernel (algebra)|kernel]] of Ψ is the smallest closed subspace containing {0}.</ref>
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| Second, even in the locally convex setting, several natural vector space topologies can be defined on the continuous dual {{nowrap|''V′''}}, so that the continuous double dual {{nowrap|''V′′''}} is not uniquely defined as a set. Saying that Ψ maps from ''V'' to {{nowrap|''V′′''}}, or in other words, that Ψ(''x'') is continuous on {{nowrap|''V′''}} for every {{nowrap|''x'' ∈ ''V''}}, is a reasonable minimal requirement on the topology of {{nowrap|''V′''}}, namely that the evaluation mappings
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| : <math>
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| \varphi \in V' \mapsto \varphi(x), \quad x \in V, \,
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| </math>
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| be continuous for the chosen topology on {{nowrap|''V′''}}. Further, there is still a choice of a topology on {{nowrap|''V′′''}}, and continuity of Ψ depends upon this choice. As a consequence, defining [[Reflexive space#Locally convex spaces|reflexivity]] in this framework is more involved than in the normed case.
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| == See also ==
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| *[[Duality (mathematics)]]
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| *[[Duality (projective geometry)]]
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| *[[Pontryagin duality]]
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| *[[Reciprocal lattice]] – dual space basis, in crystallography
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| *[[Covariance and contravariance of vectors]]
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| *[[Dual norm]]
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| == Notes ==
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| {{reflist}}
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| == References ==
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| * {{citation|authorlink = Nicolas Bourbaki|last=Bourbaki|first=Nicolas| title = Elements of mathematics, Algebra I| publisher = Springer-Verlag | year = 1989|isbn=3-540-64243-9}}
| |
| * {{citation
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| | authorlink = Nicolas Bourbaki
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| | last = Bourbaki
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| | first = Nicolas.
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| | year = 2003
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| | title = Elements of mathematics, Topological vector spaces
| |
| | publisher = Springer-Verlag
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| | location =
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| | isbn =
| |
| }}
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| * {{citation|authorlink=Paul Halmos|first=Paul|last=Halmos|title=Finite-dimensional Vector Spaces|year=1974|publisher=Springer|isbn=0-387-90093-4}}
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| * {{Lang Algebra|edition=3r}}
| |
| * {{citation|first1=Saunders|last1=MacLane|authorlink1=Saunders MacLane|first2=Garrett|last2=Birkhoff|authorlink2=Garrett Birkhoff|title=Algebra|edition=3rd|publisher=AMS Chelsea Publishing|year=1999|isbn=0-8218-1646-2}}.
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| * {{Citation
| |
| | first1=Charles W.
| |
| | last1=Misner
| |
| | authorlink1=Charles W. Misner
| |
| | first2=Kip S.
| |
| | last2=Thorne
| |
| | authorlink2=Kip S. Thorne
| |
| | first3=John A.
| |
| | last3=Wheeler
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| | authorlink3=John Archibald Wheeler
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| | title=Gravitation
| |
| | publisher= W. H. Freeman
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| | year=1973
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| | isbn=0-7167-0344-0
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| }}
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| * {{Cite isbn|9780070542365}} <!-- Rudin's Functional Analysis -->
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| * {{cite book
| |
| | last1 = Robertson
| |
| | first1 = A.P.
| |
| | last2 = Robertson
| |
| | first2 = W.
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| <!-- | authorlink = A.P.Robertson, W.Robertson -->
| |
| | year = 1964
| |
| | title = Topological vector spaces
| |
| | series=
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| | volume=
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| | publisher = Cambridge University Press
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| | location =
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| | isbn =
| |
| }}
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| * {{cite book
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| | last = Schaefer
| |
| | first = Helmuth H.
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| <!-- | authorlink = Helmuth Schaefer -->
| |
| | year = 1971
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| | title = Topological vector spaces
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| | series=[[Graduate Texts in Mathematics|GTM]]
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| | volume=3
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| | publisher = Springer-Verlag
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| | location = New York
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| | isbn = 0-387-98726-6
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| }}
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| {{linear algebra}}
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| {{Functional Analysis}}
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| {{DEFAULTSORT:Dual Space}}
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| [[Category:Linear algebra]]
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| [[Category:Functional analysis]]
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| [[Category:Duality theories|Space]]
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