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Suppose $X$ is a complex normed space of dimension 2 or 3 and $X$ is isometrically isomorphic to its dual. Is $X$ a Hilbert space? Remarks: There are easy counterexamples in the real case, and in hi …

There is active interest in such results in high-dimensional geometry, and expander graphs have even been used explicitly as a tool. Take a look for example at this paper and the references on the se …

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 function …

The inequality can't be true without additional assumptions. To see this, let $X = \ell_2^n$ and let $x$ have a spherically symmetric distribution and let $R = \Vert x \Vert$. Then $R$ is an essenti …

In Ledoux and Talagrand's "Probability in Banach Spaces", for technical reasons they frequently assume that a Banach space $B$ has the property that the unit ball of $B^*$ contains a countable subset …

I assume you also want your compact sets to have non-empty interior, hence positive volume. The literature mostly deals with the related Banach-Mazur metric $d_{BM}(A,B)$, in which it is assumed that …

Somehow I didn't register how strong the assumptions Tom was making were, hence the fact that my other answer missed the point. Unless I'm still missing something, this is very easy. Say $Z$ is a Gau …

In general, a sequence of Banach space-valued random variables $Y_n$ converges weakly to $Y$ if $f(Y_n)\to f(Y)$ for every $f\in X^*$, and $Y_n$ is tight in the sense that for each $\varepsilon > 0$ t …

As pointed out in the comments, there are many Banach tensor products, but there is indeed at least one which works nicely for $L^p\otimes L^p$. In general, the algebraic tensor product $X\otimes Y^* …

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 …