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This page concerns the reflexivity of a Banach space. For Paul Halmos' notion of the reflexivity of an operator algebra or a subspace lattice , see reflexive operator algebra.

In functional analysis, a Banach space is called reflexive if it satisfies a certain abstract property involving dual spaces. Reflexive spaces turn out to have desirable geometric properties.

1 Definition

Suppose X is a Banach space. We denote by X' its continuous dual, i.e. the space of all continuous linear maps from X to the base field ( R or C). This is again a Banach space, as explained in the dual space article. So we can form the double dual X", the continuous dual of X'. There is a natural continuous linear transformation

J : X → X"

defined by

J(x)(φ) = φ(x)     for every x in X and φ in X'.

As a consequence of the Hahn-Banach theorem, J is norm-preserving (i.e., ||J(x)||=||x|| ) and hence injective. The space X is called reflexive if J is bijective.

2 Examples

All Hilbert spaceIn mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. Hilbert spaces serve to clarify and generalize the concept of Fourier expansion, certain linear transformations such as the Fs are reflexive, as are the L^p spacesIn mathematics, the Lp and spaces are spaces of p-power integrable functions and corresponding sequence spaces''. They form an important class of examples of Banach spaces in functional analysis, and of topological vector spaces. See also root mean square for 1 < p < ∞. More generally: all uniformly convex Banach spaces are reflexive according to the Milman-Pettis theorem .

3 Properties

Every closed subspace of a reflexive space is reflexive.

The promised geometric property of reflexive spaces is the following: if C is a closed non-empty convexIn mathematics, an object is convex if for any pair of points within the object, any point on the straight line segment that joins them is also within the object. For example, a solid cube is convex, but anything that is hollow or has a dent in it is not subset of the reflexive space X, then for every x in X there exists a c in C such that ||x - c|| minimizes the distance between x and points of C. (Note that while the minimal distance between x and C is uniquely defined by x, the point c is not.)

A banach space is reflexive if and only if its dual is reflexive.

A space is reflexive if and only if its unit ball is compactIn mathematics, a compact space is a space that resembles a closed and bounded subset of Euclidean space R n in that it is "small" in a certain sense and "contains all its limit points". The modern general definition calls a topological space compact if e in the weak topologyFunctional analysis Topology General topology In mathematics, the weak topology on a set, with respect to a collection of functions from that set into topological spaces, is the weakest (that is, smallest or coarsest) topology on the set which makes all t.