What is an isomorphism of Banach spaces?

Question

The nLab page on Banach spaces (http://ncatlab.org/nlab/show/Banach%20space) was recently criticised as being, in effect, too heavily biased to category theory (not of the Baire kind) and not enough reflecting how Banach spaces are treated "in the real world" (or the closest approximation thereof that functional analysts live in). A couple of functional analysts have stepped in and are helping out, but I thought I'd also ask here just to see if there was anything more that we were missing.

So far, we have the following notions of morphism and isomorphism:

1. Morphisms are continuous (aka bounded) linear maps, isomorphisms are linear homeomorphisms (aka bi-Lipschitz linear equivalences).

2. Morphisms are "short maps", aka continuous linear maps of norm at most 1, isomorphisms are isometries.

Are there any others that are in reasonably common use or do we have them all?

Answer 1

Strictly speaking, the norm of a Banach space is part of its structure, and two equivalent norms give two different Banach spaces. Since an isomorphism should preserve the whole structure, norm included, I think the answer should be 2. Answer 1 is the natural one if we want to treat Banach space up to equivalent norms, that, is topological linear space whose topology can be given by some complete norm. To solve the ambiguity, Serge Lang uses the term Banachable for the latter case - and analogously, Hilbertable (in Fundamentals of Differential Geometry). There are other meaningful classes of linear maps that make Banach spaces into a category. I'd add to the list:

3. Unbounded linear operators.

4. Bounded Fredholm linear operators (the corresponding category of differential manifolds is a natural setting for the Fredholm Degree Theory and Orientability).

Answer 2

Mark gives a good answer. I thought to make this a comment on his answer, but I rambled on past the allowed length and so post as an answer.

It is all a matter of what maps one wants to study. As Mark noted, most analysts as well as probabilists are most interested in (1), the category Ban. The less flexible and easier to treat category Ban$_1$, given by (2), is interesting for many people and was the first to be developed to a high degree. PDE people, interested in Lipschitz mappings, naturally are care about biLipschitz equivalence: some geometers like uniform equivalence; while geometric group theorists are mostly interested in coarse equivalence. I am interested in all of these notions of equivalence and more.

No matter what category one works in, the word isomorphism means linear homeomorphism (usually into; one adds onto or surjective when called for). Other notions of being the same are called isometric, Lipschitz equivalent, uniformly equivalent, coarsely equivalent. For a time, geometric group theorists called "coarse equivalence" "uniform equivalence", but this fortunately is passing.

From a Banach space theoretic perspective, one major challenge is to determine when a weaker notion of equivalence (or embedding) implies isomorphic equivalence (or isomorphic embedding). This is interesting also for people who use Banach spaces without doing Banach space theory. Take, for example, geometric group theorists. Yu and then Kasparov and Yu proved numerous results about finitely generated groups whose Cayley graphs coarsley embed into a "nice" Banach space. For a time it was open whether every "nice" (in this case uniformly convex) Banach space embeds into a Hilbert space--were this true, they could have ignored other Banach spaces. Alas (or YES!, depending on your point of view), that is not the case. It is now a research topic of interest to a large group to determine when a Banach spaces embeds coarsely into a special Banach space $X$. For $X$ a Hilbert space (or, more generally, $L_p$ and $\ell_p$ for $p \le 2$), the answer was provided by Nirina Randrianarivony, but there are only partial results for $L_p$ when $2<p<\infty$.

Answer 3

A variation of 2. is to let morphisms be isometries into, so that isomorphisms are surjective isometries.

The other categories that I have alluded to elsewhere are those studied in nonlinear functional analysis. Namely, one may take morphisms to be Lipschitz or uniformly continuous nonlinear maps (by which I of course really mean not-necessarily-linear maps). The Lipschitz and uniform classification of Banach spaces have a very rich literature, which I am sadly mostly ignorant about (you should try to lure Bill Johnson into telling you more about it). The standard reference is Geometric Nonlinear Functional Analysis by Benyamini and Lindenstrauss.

Incidentally, there's also the question (which I asked here some time ago) about what categories could provide a good framework for understanding finite-dimensional Banach space theory, in which one often distinguishes between "isometric", "isomorphic", and "almost isometric" results.

Added: It's probably instructive to compare this discussion with the answers to Andrew's earlier question about morphisms between metric spaces, particularly Greg Kuperberg's answer.

Also, to add some context to the discussion of nonlinear maps between Banach spaces, recall that a linear map is continuous iff it is uniformly continuous iff it is Lipschitz; removing linearity means one can choose many different levels of topological/metric structure to preserve. The two examples I gave above are, I think, the ones of most interest. The two opposite extremes are of less interest for good reason. On the one hand, an old result of Mazur and Ulam shows that an isometry of a Banach space onto another Banach space is necessarily affine, so the (iso)metric structure of a Banach space already encodes its affine structure. On the other hand, a much harder theorem of Kadec shows that all separable infinite-dimensional Banach spaces are homeomorphic, so the nonlinear topological category of Banach spaces is not very interesting at all. Again, see the B&L book for more.