Throughout, we work in the category 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 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 and . is the left adjoint of , that is
so that is not a right adjoint of . For example: take to be the ground field and to be with the usual supremum norm.
Not-a-proof-yet of this claim: we have and the -morphisms from to are just vectors in ; but the -morphisms from to correspond to the vectors in . (It would seem from this example that even in dream mathematics one doesn’t get being a right adjoin of .)
To get things to run smoothly, we seem to need more than being monic in ; but I (YC) am not sure which of the usual variants – extreme, regular, strong, strict – is the key one.
The counit map 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 is reflexive if is an isomorphism in . If we furthermore grant ourselves Hahn-Banach, then 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 .
It turns out that if is a Banach space, then it is reflexive if and only if its (norm-)closed unit ball is compact in the -topology. In particular, if is reflexive and , are bounded linear operators, then the composition 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.
In dream mathematics is reflexive; but its closed subspace is not. On the other hand, if one works in a setting where 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 . More is true: is a codimension- subspace of .
One can renorm to obtain a non-reflexive Banach space which is isometrically isomorphic as a Banach space to its second dual (James, 1951).
MR0044024 (13,356d) James, Robert C. A non-reflexive Banach space isometric with its second conjugate space. Proc. Nat. Acad. Sci. U. S. A. 37, (1951). 174–177.
MR0355536 (50 #8010) Davis, W. J.; Figiel, T.; Johnson, W. B.; Pełczyński, A. Factoring weakly compact operators. J. Functional Analysis 17 (1974), 311–327.