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Your inequality is implicit in Hurwitz's Fourier series proof of the isoperimetric inequality in the plane. See for example section 36 of Körner's Fourier Analysis or section 4.1 of Groemer's Geometr …

Although the construction of Tsirelson's space doesn't use set theory per se, in this short essay Tsirelson recounts (among other things) how his construction was inspired by forcing.

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 …

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 …

Suppose $H$ is the reproducing kernel Hilbert space on a space $X$ with reproducing kernel $K$. If, say, $K - c$ is a positive definite kernel for some $c>0$ then $H$ contains the constant functions …

I'm not sure exactly what you have in mind by "need the Hahn-Banach theorem". One standard example of something pretty concrete for which Hahn-Banach in some form is needed is to show that there are …

A popular pair of exercises in first courses on functional analysis prove the following theorem: The unit ball of a Banach space $X$ is compact if and only if $X$ is finite-dimensional. My quest …

The second example in the original post generalizes a lot. Let $d_i$ be finitely or countably many pseudometrics (it's possible for $d_i(x,y)=0$ even if $x\neq y$) for $i\ge 1$, and assume $d_1$ is a …

The distributions you're looking for are stable distributions. Basically, the only such norms you can take are $\ell^p$ norms for $1 \le p \le 2$. If you don't need an honest norm, you can also take …

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 Nate and I pointed out in comments, your question reduces to asking whether there is a unified framework which includes both $L^p$ spaces and Schatten spaces. One such framework (there may be othe …

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 …

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 …

Since you read German, my favorite is Funktionalanalysis by Dirk Werner. It's not necessarily comprehensive, but it covers a lot, has extensive historical remarks, and is extremely well-written -- I …

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 …