Dual Space - Continuous Dual Space - Double Dual

Double Dual

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

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

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 xV, 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 xV, is a reasonable minimal requirement on the topology of V′, namely that the evaluation mappings

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

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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