Editing Dual space (section)

=== 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.