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Concept
Reflexive space
Summary
In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space (TVS) for which the canonical evaluation map from into its bidual (which is the strong dual of the strong dual of ) is an isomorphism of TVSs.
Since a normable TVS is reflexive if and only if it is semi-reflexive, every normed space (and so in particular, every Banach space) is reflexive if and only if the canonical evaluation map from into its bidual is surjective;
in this case the normed space is necessarily also a Banach space.
In 1951, R. C. James discovered a Banach space, now known as James' space, that is reflexive but is nevertheless isometrically isomorphic to its bidual (any such isomorphism is thus necessarily the canonical evaluation map).
Reflexive spaces play an important role in the general theory of locally convex TVSs and in the theory of Banach spaces in particular. Hilbert spaces are prominent examples of reflexive Banach spaces. Reflexive Banach spaces are often characterized by their geometric properties.
Definition of the bidual
Bidual
Suppose that is a topological vector space (TVS) over the field (which is either the real or complex numbers) whose continuous dual space, separates points on (that is, for any there exists some such that ).
Let and both denote the strong dual of which is the vector space of continuous linear functionals on endowed with the topology of uniform convergence on bounded subsets of ;
this topology is also called the strong dual topology and it is the "default" topology placed on a continuous dual space (unless another topology is specified).
If is a normed space, then the strong dual of is the continuous dual space with its usual norm topology.
The bidual of denoted by is the strong dual of ; that is, it is the space
If is a normed space, then is the continuous dual space of the Banach space with its usual norm topology.
Official source
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