Reflexive space
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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.
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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'.
That is, J maps x to the functional on X' given by evaluation at 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.
Examples
All Hilbert spaces are reflexive, as are the Lp spaces for 1 < p < ∞. More generally: all uniformly convex Banach spaces are reflexive according to the Milman-Pettis theorem.
Montel spaces are reflexive.
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 convex 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 compact in the weak topology.
Implications
A reflexive space is separable if and only if its dual is separable.
If a space is reflexive, then every bounded sequence has a weakly convergent subsequence.
A Banach space X is reflexive iff each linear bounded functional defined in X attains its maximum on the unit ball of X (Famous James Theorem).de:Reflexivität