2
votes
0answers
98 views

Projective tensor powers of Banach spaces over a normed field

Let $E$ be a Banach space over a complete normed field $\mathbb K$. Is it possible to classify all proper $E$ for which the projective seminorm $p_n$ defined on the $n$-th tensor power of $E$ is a ...
6
votes
0answers
214 views

Is there Ultracoproduct-like construction for topological spaces in general?

In http://arxiv.org/pdf/math/9704205.pdf they define the ultracoproduct of a sequence of compact Hausdorff spaces, $\sum_\mathcal{U}X_i$ along an ultrafilter $\mathcal{U}$ as the Wallman-Frink ...
10
votes
3answers
2k views

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 ...
19
votes
2answers
972 views

What theorem constructs an initial object for this category? (Formerly “Integrability by abstract nonsense”)

Tom Leinster has a note here about how you can realize L^1[0,1] as the initial object of a certain category. You should really read his note because it is only 2.5 pages and is much more charming than ...
3
votes
2answers
350 views

Which Banach spaces have categorical duals?

I was looking carefully at all the definitions, trying to understand exactly what was going on in this question on categorical duals in Banach spaces. It seems that in the category of Banach spaces ...
8
votes
4answers
869 views

Is the category of Banach spaces with contractions an algebraic theory?

Consider the category of Banach spaces with contractions as morphisms (weak, so $\|T\| \le 1$). Is this an algebraic theory? I suspect that this is true. The "operations" will be weighted sums, ...
11
votes
3answers
604 views

What are the right categories of finite-dimensional Banach spaces?

This is inspired partly by this question, especially Tom Leinster's answer. Let me start with some background. I apologize that this will be rather long, since I'm hoping for input from people who ...