# What is an isomorphism of Banach spaces?

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?

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I too know only these two notions. For other topics about BAnach spaces category study I guess you just know the book: Johann Cigler, Viktor Losert, Peter W. Michor: Banach modules and functors on categories of Banach spaces (see in mat.univie.ac.at/~michor/listpubl.html) – Buschi Sergio Nov 10 '11 at 8:25
I apologize for this blatant self-advertisement, but I've collected a number of basic categorical properties of the two categories you mention in chapter IV of my thesis. Most of this is contained in Cigler-Losert-Michor, but there are some aspects that aren't: dx.doi.org/10.3929/ethz-a-005561107 – Theo Buehler Nov 10 '11 at 11:10
No need to apologize Theo; on the contrary, I'm glad to learn of this. We may wind up linking to it in an nLab article! – Todd Trimble Nov 10 '11 at 12:18
@Andreas: the issue is, which subject are you inventing? If you'd just invented Banach spaces out of thin air, version 2 makes sense. But if you'd invented Banach space theory as a tool to help solve other problems in analysis (which is what Banach and his group actually did), you'd probably have found that version 1 was usually more relevant to those problems. – Mark Meckes Nov 10 '11 at 14:28
I should point out that the category (2) is very natural from the nPOV (ncatlab.org/nlab/show/nPOV), in fact it is an essential ingredient in the correct formuation (from the nPOV) of the Hahn-Banach theorem. Namely, there is a contravariant equivalence from the category (2) to the appropriately defined category of unit balls, which sends a Banach space to the unit ball of its dual space equipped with the weak topology. This variant of the Hahn-Banach theorem is valid for locales and can be proved without using any form of the axiom of choice. – Dmitri Pavlov Nov 10 '11 at 19:55

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).

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Yes, it did come up in our discussion that 1. is more about Banachable spaces. Somehow I don't see that name catching on! – Loop Space Nov 10 '11 at 9:31
Pietro, I'm not knowledgeable about unbounded linear operators. If it is usual to require that the domains of unbounded linear operators $E \to F$ are dense in $E$, then is it clear that unbounded linear operators compose? Is there a reference for this? – Todd Trimble Nov 10 '11 at 12:47
Todd: no, they need not. The domain of one operator need not contain the range of another. In fact their intersection can be trivial. So this is probably not a good notion from your point of view. – Nate Eldredge Nov 10 '11 at 13:29
Thanks, Nate; that's what I figured. By the way, I disown "my point of view", because I think it's really what the post itself is about: what categories whose objects are Banach spaces do people use? – Todd Trimble Nov 10 '11 at 14:24
As to unbounded operator, the idea is to define the domain of the composition as $\operatorname{dom}(TS):=S^{-1}(\operatorname{dom}(T)\subset \operatorname{dom}(S)$ (which could be trivial, of course). Naturally I know no use of this category. – Pietro Majer Nov 10 '11 at 21:36

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 $2<p<\infty$.

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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.

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