reflexive Banach space


Throughout, we work in the category BantBant whose objects are Banach spaces and where the morphisms are continuous linear maps. References below to the unit ball suggest that it might be premature to cast everything in terms of a certain subcategory of TVS.


Following the lead of Mac Lane (2nd ed.) Section IV.2, which does this for vector spaces) but with slight changes in notation: let D¯:BantBant\overline{D}:Bant\to Bant be the contravariant functor which takes a Banach space to its dual space, and sends a continuous linear map to its adjoint/dual map.

This gives rise, in a straightforward way, to two functors D L:BantBant opD_L: Bant \to Bant^{op} and D R:Bant opBantD_R: Bant^{op}\to Bant. D LD_L is the left adjoint of D RD_R, that is

Bant op(D LX,Y)Bant(X,D RY) Bant^{op}(D_L X, Y) \cong Bant(X, D_R Y)

In general

Bant op(Y,D LX)¬Bant(D RY,X) Bant^{op}(Y, D_L X) \not\cong Bant(D_R Y, X)

so that D LD_L is not a right adjoint of D RD_R. For example: take YY to be the ground field KK and XX to be c 0c_0 with the usual supremum norm.

Not-a-proof-yet of this claim: we have D R(K)KD_R(K)\cong K and the BantBant-morphisms from KK to c 0c_0 are just vectors in c 0c_0; but the BantBant-morphisms from D L(c 0) 1D_L(c_0)\cong\ell^1 to KK correspond to the vectors in \ell^\infty. (It would seem from this example that even in dream mathematics one doesn’t get D LD_L being a right adjoin of D RD_R.)

Unit and counit. The unit of this adjunction is the canonical map κ X:X(X *) *\kappa_X: X\to (X^*)^* from a Banach space XX to its second dual X **X^{**}. In the presence of Choice, the Hahn–Banach theorem ensures that κ X\kappa_X is an isometry.

To get things to run smoothly, we seem to need more than κ X\kappa_X being monic in BantBant; but I (YC) am not sure which of the usual variants – extreme, regular, strong, strict – is the key one.

The counit map veps X:X ***X *\veps_X: X^{***} \to X^* is sometimes known as the Dixmier projection from the third dual of a Banach space to (the canonical image of) its first dual; note that this map is weak-star-to-weak-star continuous. (It is a projection in the sense of vector spaces, by the triangle identity for the adjunction.)


A Banach space XX is reflexive if κ X\kappa_X is an isomorphism in BantBant. If we furthermore grant ourselves Hahn-Banach, then κ X\kappa_X will even be an isometric isomorphism: an isomorphism in the category of Banach spaces and short linear maps.



If two Banach spaces are isomorphic as TVSes (but not necessarily isometrically isomorphic), then either both are reflexive or both are non-reflexive.

There is a nice proof that closed subspaces of reflexive Banach spaces are reflexive (due to Linton? ) using naturality of κ X\kappa_X.

It turns out that if XX is a Banach space, then it is reflexive if and only if its (norm-)closed unit ball is compact in the σ(X,X *)\sigma(X,X^*)-topology. In particular, if XX is reflexive and EXE\to X, XFX\to F are bounded linear operators, then the composition EFE\to F is weakly compact? as a linear operator.

A theorem of Davis-Figiel-Johnson-Pelczynski (1974) tells us that the converse is true: every weakly compact? linear operator between Banach spaces factors through some reflexive Banach space. The intermediate space is constructed by real interpolation and (at least as usually presented) does not seem to be canonical in any way.

Examples and counterexamples

By the Riesz duality theorem?, every separable Hilbert space is reflexive.

The Lebesgue space l 1l^1 is reflexive in dream mathematics, but in classical mathematics it is not.

In dream mathematics l l^\infty is reflexive; but its closed subspace c 0c_0 is not. On the other hand, if one works in a setting where κ X\kappa_X is a monomorphism, then closed subspaces of reflexive Banach spaces are reflexive (see above)

James (1950) constructed a separable non-reflexive Banach space which is isomorphic as a TVS to its second dual; nowadays this is known as the James space JJ. More is true: κ J(J)J\kappa_J(J)J is a codimension-11 subspace of J **J^{**}.

One can renorm JJ to obtain a non-reflexive Banach space which is isometrically isomorphic as a Banach space to its second dual (James, 1951).