**Double Dual**

In analogy with the case of the algebraic double dual, there is always a naturally defined continuous linear operator Ψ : *V* → *V′′* from a normed space *V* into its continuous double dual *V′′*, defined by

As a consequence of the Hahn–Banach theorem, this map is in fact an isometry, meaning ||Ψ(*x*)|| = ||*x*|| for all *x* in *V*. Normed spaces for which the map Ψ is a bijection are called reflexive.

When *V* is a topological vector space, one can still define Ψ(*x*) by the same formula, for every *x* ∈ *V*, however several difficulties arise. First, when *V* is not locally convex, the continuous dual may be equal to {0} and the map Ψ trivial. However, if *V* is Hausdorff and locally convex, the map Ψ is injective from *V* to the algebraic dual *V′** of the continuous dual, again as a consequence of the Hahn–Banach theorem.

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

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

Read more about this topic: Dual Space, Continuous Dual Space

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