Editing Dual space (section)

== Continuous dual space ==<!-- This section is linked from [[Reflexive space]] -->

When dealing with [[topological vector space]]s, the [[continuous function (topology)|continuous]] linear functionals from the space into the base field <math>\mathbb{F} = \Complex</math> (or <math>\R</math>) 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 <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 often replaced by "dual space".

For a [[topological vector space]] <math>V</math> its ''continuous dual space'',<ref name="A.P.Robertson, W.Robertson 1964 loc=II.2">{{harvnb|Robertson|Robertson|1964|loc=II.2}}</ref> or ''topological dual space'',<ref name="Schaefer 1966 loc=II.4">{{harvnb|Schaefer|1966|loc=II.4}}</ref> or just ''dual space''<ref name="A.P.Robertson, W.Robertson 1964 loc=II.2"/><ref name="Schaefer 1966 loc=II.4"/><ref>{{harvnb|Rudin|1973|loc=3.1}}</ref><ref>{{harvnb|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>.

Important examples for continuous dual spaces are the space of compactly supported [[Spaces of test functions and distributions|test functions]] <math>\mathcal{D}</math> and its dual <math>\mathcal{D}',</math> the space of arbitrary [[Distribution (mathematics)|distributions]] (generalized functions); the space of arbitrary test functions <math>\mathcal{E}</math> and its dual <math>\mathcal{E}',</math> the space of compactly supported distributions; and the space of rapidly decreasing test functions <math>\mathcal{S},</math> the [[Schwartz space]], and its dual <math>\mathcal{S}',</math> the space of [[Distribution (mathematics)#Tempered distributions and Fourier transform|tempered distributions]] (slowly growing distributions) in the theory of [[generalized function]]s.

=== Properties ===

If {{mvar|X}} is a [[Hausdorff space|Hausdorff]] [[topological vector space]] (TVS), then the continuous dual space of {{mvar|X}} is identical to the continuous dual space of the [[Complete topological vector space|completion]] of {{mvar|X}}.{{sfn | Narici|Beckenstein | 2011 | pp=225-273}}

=== Topologies on the dual ===
{{Main|Polar topology|Dual system}}

There is a standard construction for introducing a topology on the continuous dual <math>V'</math> of a topological vector space <math>V</math>. Fix a collection <math>\mathcal{A}</math> of [[Bounded set (topological vector space)|bounded subsets]] of <math>V</math>.
This gives the topology on <math>V</math> of uniform convergence on sets from <math>\mathcal{A},</math> or what is the same thing, the topology generated by [[seminorm]]s of the form

:<math>\|\varphi\|_A = \sup_{x\in A} |\varphi(x)|,</math>

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>

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

:<math>\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. </math>

Usually (but not necessarily) the class <math>\mathcal{A}</math> is supposed to satisfy the following conditions:
* Each point <math>x</math> of <math>V</math> belongs to some set <math>A\in\mathcal{A}</math>:
*:<math>\text{ for all } x \in V\quad \text{ there exists some } A \in \mathcal{A}\quad \text{ such that } x \in A.</math>
* 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>:
*:<math>\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.</math>
* <math>\mathcal{A}</math> is closed under the operation of multiplication by scalars: 
*:<math>\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}.</math>

If these requirements are fulfilled then the corresponding topology on <math>V'</math> is Hausdorff and the sets

:<math>U_A ~=~ \left \{ \varphi \in V' ~:~ \quad \|\varphi\|_A < 1 \right \},\qquad \text{ for } A \in \mathcal{A}</math>

form its local base.

Here are the three most important special cases.
* 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]] (for example, 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
::<math>\|\varphi\| = \sup_{\|x\| \le 1 } |\varphi(x)|.</math>
* 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>).
* 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>).

Each of these three choices of topology on <math>V'</math> leads to a variant of [[Reflexive space|reflexivity property]] for topological vector spaces:
* If <math>V'</math> is endowed with the [[strong topology]], then the corresponding notion of reflexivity is the standard one: the spaces reflexive in this sense are just called ''reflexive''.<ref name="H.Schaefer 1966 loc=IV.5.5">{{harvnb|Schaefer|1966|loc=IV.5.5}}</ref>
* If <math>V'</math> is endowed with the stereotype dual topology, then the corresponding reflexivity is presented in the theory of [[stereotype space]]s: the spaces reflexive in this sense are called ''stereotype''.
* If <math>V'</math> is endowed with the [[weak topology]], then the corresponding reflexivity is presented in the theory of [[dual pair]]s:<ref name="H.Schaefer 1966 loc=IV.1">{{harvnb|Schaefer|1966|loc=IV.1}}</ref> the spaces reflexive in this sense are arbitrary (Hausdorff) locally convex spaces with the weak topology.<ref name="H.Schaefer 1966 loc=IV.1.2">{{harvnb|Schaefer|1966|loc=IV.1.2}}</ref>

=== Examples ===

Let 1 < ''p'' < ∞ be a real number and consider the Banach space ''[[Lp space#The p-norm in countably infinite dimensions|ℓ<sup>&thinsp;p</sup>]]'' of all [[sequence]]s {{math|1='''a''' = (''a''<sub>''n''</sub>)}} for which

:<math>\|\mathbf{a}\|_p = \left ( \sum_{n=0}^\infty |a_n|^p \right) ^{\frac{1}{p}} < \infty.</math>

Define the number ''q'' by {{math|1=1/''p'' + 1/''q'' = 1}}. Then the continuous dual of ''ℓ''<sup>&thinsp;''p''</sup> is naturally identified with ''ℓ''<sup>&thinsp;''q''</sup>: given an element <math>\varphi \in (\ell^p)'</math>, the corresponding element of {{math|''ℓ''<sup>&thinsp;''q''</sup>}} is the sequence <math>(\varphi(\mathbf {e}_n))</math> where '''<math>\mathbf {e}_n</math>''' denotes the sequence whose {{mvar|n}}-th term is 1 and all others are zero. Conversely, given an element {{math|1='''a''' = (''a''<sub>''n''</sub>) ∈ ''ℓ''<sup>&thinsp;''q''</sup>}}, the corresponding continuous linear functional ''<math>\varphi</math>'' on {{math|''ℓ''<sup>&thinsp;''p''</sup>}} is defined by

:<math>\varphi (\mathbf{b}) = \sum_n a_n b_n</math>

for all {{math|1='''b''' = (''b<sub>n</sub>'') ∈ ''ℓ''<sup>&thinsp;''p''</sup>}} (see [[Hölder's inequality]]).

In a similar manner, the continuous dual of {{math|''ℓ''<sup>&thinsp;1</sup>}} is naturally identified with {{math|''ℓ''<sup>&thinsp;∞</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 {{math|''ℓ''<sup>&thinsp;1</sup>}}.

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]].

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

=== Transpose of a continuous linear map ===
{{See also|Transpose of a linear map|Dual system#Transposes}}

If {{math|''T'' : ''V → W''}} is a continuous linear map between two topological vector spaces, then the (continuous) transpose  {{math|''T′'' : ''W′ → V′''}} is defined by the same formula as before:

:<math>T'(\varphi) = \varphi \circ T, \quad \varphi \in W'.</math>

The resulting functional {{math|''T′''(''φ'')}} is in {{math|''V′''}}. The assignment {{math|''T → T′''}} produces a linear map between the space of continuous linear maps from ''V'' to ''W'' and the space of linear maps from {{math|''W′''}} to {{math|''V′''}}.
When ''T'' and ''U'' are composable continuous linear maps, then

:<math>(U \circ T)' = T' \circ U'.</math>

When ''V'' and ''W'' are normed spaces, the norm of the transpose in{{math| ''L''(''W′'', ''V′'')}} is equal to that of ''T'' in {{math|''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 {{math|''T′''}} is injective.

When ''T'' is a [[Compact operator|compact]] linear map between two Banach spaces ''V'' and ''W'', then the transpose {{math|''T′''}} is compact.
This can be proved using the [[Arzelà–Ascoli theorem]].

When ''V'' is a Hilbert space, there is an antilinear isomorphism ''i<sub>V</sub>'' from ''V'' onto its continuous dual {{math|''V′''}}.
For every bounded linear map ''T'' on ''V'', the transpose and the [[Hermitian adjoint|adjoint]] operators are linked by

:<math>i_V \circ T^* = T' \circ i_V.</math>

When ''T'' is a continuous linear map between two topological vector spaces ''V'' and ''W'', then the transpose {{math|''T′''}} is continuous when {{math|''W′''}} and {{math|''V′''}} are equipped with "compatible" topologies: for example, when for {{math|1=''X'' = ''V''}} and {{math|1=''X'' = ''W''}}, both duals {{math|''X′''}} have the [[Strong topology (polar topology)|strong topology]] {{math|''β''(''X′'', ''X'')}} of uniform convergence on bounded sets of ''X'', or both have the weak-∗ topology {{math|''σ''(''X′'', ''X'')}} of pointwise convergence on&nbsp;''X''.
The transpose {{math|''T′''}} is continuous from {{math|''β''(''W′'', ''W'')}} to {{math|''β''(''V′'', ''V'')}}, or from {{math|''σ''(''W′'', ''W'')}} to {{math|''σ''(''V′'', ''V'')}}.

=== Annihilators ===

Assume that ''W'' is a closed linear subspace of a normed space&nbsp;''V'', and consider the annihilator of ''W'' in {{math|''V′''}},

:<math>W^\perp = \{ \varphi \in V' : W \subseteq \ker \varphi\}.</math>

Then, the dual of the quotient {{math|''V''&thinsp;/&thinsp;''W''&thinsp;}} can be identified with ''W''<sup>⊥</sup>, and the dual of ''W'' can be identified with the quotient {{math|''V′''&thinsp;/&thinsp;''W''<sup>⊥</sup>}}.<ref>{{harvnb|Rudin|1991|loc=chapter 4}}</ref>
Indeed, let ''P'' denote the canonical [[surjection]] from ''V'' onto the quotient {{math|''V''&thinsp;/&thinsp;''W''&thinsp;}}; then, the transpose {{math|''P′''}} is an isometric isomorphism from {{math|(''V''&thinsp;/&thinsp;''W''&thinsp;)′}} into {{math|''V′''}}, with range equal to ''W''<sup>⊥</sup>.
If ''j'' denotes the injection map from ''W'' into ''V'', then the kernel of the transpose {{math|''j′''}} is the annihilator of ''W'':
:<math>\ker (j') = W^\perp</math>
and it follows from the [[Hahn–Banach theorem]] that {{math|''j′''}} induces an isometric isomorphism
{{math|''V′''&thinsp;/&thinsp;''W''<sup>⊥</sup> → ''W′''}}.

=== Further properties ===

If the dual of a normed space {{mvar|V}} is [[separable space|separable]], then so is the space {{mvar|V}} itself.
The converse is not true: for example, the space {{math|''ℓ''<sup>&thinsp;1</sup>}} is separable, but its dual {{math|''ℓ''<sup>&thinsp;∞</sup>}} is not.

=== Double dual ===

[[File:Double dual nature.svg|thumbnail|This is a [[natural transformation]] of vector addition from a vector space to its double dual. {{math|{{langle}}''x''<sub>1</sub>, ''x''<sub>2</sub>{{rangle}}}} denotes the [[ordered pair]] of two vectors. The addition + sends ''x''<sub>1</sub> and ''x''<sub>2</sub> to {{math|''x''<sub>1</sub> + ''x''<sub>2</sub>}}.
The addition +′ induced by the transformation can be defined as ''<math>[\Psi(x_1) +' \Psi(x_2)](\varphi) = \varphi(x_1 + x_2) = \varphi(x)</math>'' for any ''<math>\varphi</math>'' in the dual space.]]
In analogy with the case of the algebraic double dual, there is always a naturally defined continuous linear operator {{math|Ψ : ''V'' → ''V′′''}} from a normed space ''V'' into its continuous double dual {{math|''V′′''}}, defined by

:<math> \Psi(x)(\varphi) = \varphi(x), \quad x \in V, \ \varphi \in V' .</math>

As a consequence of the [[Hahn–Banach theorem]], this map is in fact an [[isometry]], meaning {{math|1=‖ Ψ(''x'') ‖ = ‖ ''x'' ‖}} for all {{math|''x'' ∈ ''V''}}.
Normed spaces for which the map Ψ is a [[bijection]] are called [[reflexive space|reflexive]].

When ''V'' is a [[topological vector space]] then Ψ(''x'') can still be defined by the same formula, for every {{math|''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 {{math|''V′''<sup>∗</sup>}} of the continuous dual, again as a consequence of the Hahn–Banach theorem.<ref group=nb>If ''V'' is locally convex but not Hausdorff, the [[kernel (algebra)|kernel]] of Ψ is the smallest closed subspace containing {0}.</ref>

Second, even in the locally convex setting, several natural vector space topologies can be defined on the continuous dual {{math|''V′''}}, so that the continuous double dual {{math|''V′′''}} is not uniquely defined as a set. Saying that Ψ maps from ''V'' to {{math|''V′′''}}, or in other words, that Ψ(''x'') is continuous on {{math|''V′''}} for every {{math|''x'' ∈ ''V''}}, is a reasonable minimal requirement on the topology of {{math|''V′''}}, namely that the evaluation mappings

: <math> \varphi \in V' \mapsto \varphi(x), \quad x \in V , </math>

be continuous for the chosen topology on {{math|''V′''}}. Further, there is still a choice of a topology on {{math|''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.