As I continue teaching Functional Analysis, I am happily learning new things, which typically come out of trying to think how to present the material in a way that motivates it for my students. I try to insert, at least at particularly important places, some example or indication that will give some understanding of why various steps (or constructions, or definitions) can be seen as natural ones. Often, this leads to natural questions which are not always explicitly mentioned or highlighted.
Here is one such situation: since I have started by treating Hilbert spaces, I try to present analogies and contrasts when it comes to Banach spaces. One such contrast, of course, involves the fact that Banach spaces are not always reflexive. What this means is that the natural (bi)duality map
is not always surjective, where D is the linear map between a Banach space V and the dual of its dual space V”, given by defining D(v) as the linear functional
for all λ in the dual space.
For a Hilbert space V, the (or one of the) Riesz Representation Theorem gives essentially an identification of V with its dual (or rather the “conjugate” Hilbert space in the case of complex scalars), from which the surjectivity follows because, properly interpreted, D is simply the identity.
To start explaining this, and why it may fail, and what “reflexivity” means, I follow the following path, which is (hopefully) quite natural here, since it is based on the Hilbert space proof, but which turned out to not be really emphasized in the textbooks I am using as references.
First, the Riesz representation theorem is typically proved (and I proved it) using the general result of existence of a projection of a vector on a closed convex set, in particular on a closed linear subspace: given W a closed subspace of V, and a vector v in V, there exists a unique w such that
and in particular the infimum on the right is attained. For a general Banach space, this result is not always true, both in terms of unicity, and of existence. I will now concentrate on the latter property (see this earlier post for an example and another nice result involving this type of minimization problems).
I used the question of extending this type of results as a first motivating fact for the Hahn-Banach theorem: one of the first corollaries of it is that, for any vector v, we have
where λ ranges over all elements of the dual space which have norm at most 1, i.e., such that
for all vectors x. The content is first that there is equality with a sup on the right-hand side, but in addition that this supremum is attained.
It is now perfectly natural to wonder whether the “dual” identity
holds, or in other words, whether a linear functional λ always attains its supremum on the closed unit ball of V (since the supremum is exactly ||λ|| by definition of the norm on the dual space).
That these are maxima instead of minima as in the previous case of Hilbert spaces is not particularly problematic, since one can always go from one to the other; in fact the counterexample mentioned earlier was exactly of this type: for a certain linear form, (**) was not correct, and a counterexample to the minimization problem followed straightforwardly.
However, it is very easy to see that (*) implies that (**) is true, for all linear forms on V, provided V is reflexive. So we get from the previous example a very straightforward proof, based on trying to understand the contrast between Hilbert and Banach spaces, that the space c0 that occured is not reflexive.
But then one can ask (and I asked myself): is the converse to this implication true? In other words, assuming we have the nice property (**) for all continuous linear maps on V, does it follow that V is reflexive?
It turns out the answer is indeed Yes, but it seems to be a difficult result of Banach space theory. This is a theorem of R.C. James, and I found it mentioned (in a slightly disguised form) in Conway’s book on functional analysis, and following the reference, found the proof contained in this paper (Studia Mathematica, 1964). It is, indeed, by no means straightforward (though I am certainly not expert enough to really understand the proof after a first glance…).
None of the other texts I am looking at for my course mentions this (unless it escaped my notice, which is quite possible). Presumably, this is because it is not a particularly fruitful approach to further studies of Banach spaces (as the unbalance between both sides of the implication suggests), but it is definitely nice to know and to mention to students.
(One of my teachers once said — and I’m sure this has been said many times of many subjects — that “completely general facts in topology are either obvious or false”; this result indicates that the same clearly cannot be said of Banach spaces).