In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space for which the canonical evaluation map from
In 1951, R. C. James discovered a Banach space, now known as James' space, that is not reflexive (meaning that the canonical evaluation map is not an isomorphism) but is nevertheless isometrically isomorphic to its bidual (any such isometric isomorphism is necessarily not the canonical evaluation map).
So importantly, for a Banach space to be reflexive, it is not enough for it to be isometrically isomorphic to its bidual; it is the canonical evaluation map in particular that has to be a homeomorphism.
Reflexive spaces play an important role in the general theory of locally convex TVSs and in the theory of Banach spaces in particular.
Reflexive Banach spaces are often characterized by their geometric properties.
is a topological vector space (TVS) over the field
(which is either the real or complex numbers) whose continuous dual space,
endowed with the topology of uniform convergence on bounded subsets of
[1] A normable space is reflexive if and only if it is semi-reflexive or equivalently, if and only if the evaluation map is surjective.
is reflexive if it is linearly isometric to its bidual under this canonical embedding
Since norm-closed convex subsets in a Banach space are weakly closed,[10] it follows from the third property that closed bounded convex subsets of a reflexive space
Thus, for every decreasing sequence of non-empty closed bounded convex subsets of
The promised geometric property of reflexive Banach spaces is the following: if
is a closed non-empty convex subset of the reflexive space
[13] One of James' characterizations of super-reflexivity uses the growth of separated trees.
Using the tree-characterization, Enflo proved[16] that super-reflexive Banach spaces admit an equivalent uniformly convex norm.
Trees in a Banach space are a special instance of vector-valued martingales.
Adding techniques from scalar martingale theory, Pisier improved Enflo's result by showing[17] that a super-reflexive space
that is,, the topology of uniform convergence on bounded subsets in
is called Theorem[19] — A locally convex Hausdorff space
-topology has the Heine–Borel property (i.e. weakly closed and bounded subsets of
Theorem[22] — The strong dual of a semireflexive space is barrelled.
is a Hausdorff locally convex space then the canonical injection from
is a normed space then the following are equivalent: Theorem[29] — A real Banach space is reflexive if and only if every pair of non-empty disjoint closed convex subsets, one of which is bounded, can be strictly separated by a hyperplane.
[23] If a barreled locally convex Hausdorff space is semireflexive then it is reflexive.
[26] A locally convex Hausdorff reflexive space is barrelled.
is called polar reflexive[34] or stereotype if the evaluation map into the second dual space
endowed with the topology of uniform convergence on totally bounded sets in
The category Ste have applications in duality theory for non-commutative groups.
– the spaces defined by the corresponding reflexivity condition are called reflective,[35][36] and they form an even wider class than Ste, but it is not clear (2012), whether this class forms a category with properties similar to those of Ste.