Reflexive space

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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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[edit] Definition

Suppose X is a normed vector space over R or C. We denote by X' its continuous dual, i.e. the space of all continuous linear maps from X to the base field. As explained in the dual space article, X' is a Banach space. 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.

Note: the definition implies all reflexive spaces are Banach spaces, since X must be isomorphic to X''.

[edit] 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. The L1 and L∞ spaces are not reflexive.

Montel spaces are reflexive.

[edit] 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.[1]

[edit] 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 consequence of the Banach-Alaoglu theorem.

[edit] See also

[edit] Notes

  1. ^ Conway, Theorem V.4.2, p.135.

[edit] References

  • J.B. Conway, A Course in Functional Analysis, Springer, 1985.