- Reflexive space
In
functional analysis , aBanach space is called reflexive if it satisfies a certain abstract property involvingdual space s. Reflexive spaces turn out to have desirable geometric properties.Definition
Suppose "X" is a
normed vector space over R or C. We denote by "X"' its continuous dual, i.e. the space of allcontinuous linear map s 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 naturalcontinuous 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 theHahn–Banach theorem , "J" is norm-preserving (i.e., ||"J"("x")|| = ||"x"|| ) and henceinjective . The space "X" is called reflexive if "J" isbijective .Note: the definition implies all reflexive spaces are Banach spaces, since "X" must be isomorphic to "X"
" .Examples
All
Hilbert space s are reflexive, as are the L"p" spaces for 1 < "p" < ∞. More generally: all uniformly convex Banach spaces are reflexive according to theMilman–Pettis theorem . The L1 and L∞ spaces are not reflexive.Montel space s 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 . [Conway, Theorem V.4.2, p.135.]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 .ee also
*
James' theorem provides a characterization of reflexive space
* A generalization which has some of the properties of reflective space and includes many space of practical importance isGrothendieck space
*Reflexive operator algebra Notes
References
*
J.B. Conway , "A Course in Functional Analysis", Springer, 1985.
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