Bill Lawvere wrote in part:
Students might wonder why contiuous linear operators are traditionally called "bounded" (when they are not even).
Then Jeff Egger wrote in part:
there's certainly alot that can be said about the category of Banach spaces and linear contractions
and:
forgetful functors don't have to be "the obvious thing";
Indeed, the "obvious" forgetful functor from Ban to Set (or Top or Met) takes a Banach space to its space of all points, while the "good" one takes the space to its unit ball. Anyway, if you mix these, then a linear transformation is bounded iff it is bounded as a function from the unit ball to the space of all points. Similarly for compact linear transformations (the image is compact). That may not be the origin of these terms, but it's how I understand them. More related to category theory itself: Jeff also wrote:
in category theory, the meaning of isomorphism is fixed
I'd say that the meaning of a term like "Banach space" necessarily includes the idea of what an isomorphism of such is. If different definitions define equivalent groupoids (or equivalent omega-groupoids in the most general case, as with different definitions of n-category, for example), then we can consider them equivalent defintions. So to define the essence of what Banach spaces are, one must specify (up to equivalence) the groupoid Ban_0 of Banach spaces and linear isometries between them. That said, there is some sense in the category Ban_b of Banach spaces and bounded linear transformations between them, but it is only a secondary notion compared to Ban_0. To be useful at all, it needs some extra structure, such as (at least) the dagger operator (giving duals of morphisms); then the actual isomorphisms of Banach spaces (those in Ban_0) are only the ~unitary~ (dual = inverse) isomorphisms in Ban_b. (In contrast, the category Ban as Jeff defined it needs no extra structure to be a sensible concept, since all of its isomorphisms are in Ban_0 already.) This dagger operator is used, for example, to make Hilb_b (the full subcategory of Ban_b whose objects are Hilbert spaces) into a 2-Hilbert space (from John Baez's HDA4), which is useful if you want examples of 2-Hilbert spaces; but the ~essence~ of what Hilbert spaces are is given by the groupoid Hilb_0 of linear isometries. So here is another lesson of category theory, to be taken together with Jeff's lesson last quoted above: Sometimes different notions of morphism are useful for different purposes. --Toby