6
votes
2answers
152 views

Local-to-global inequalities for measures: Brunn-Minkowski, Ahlswede-Daykin, what else?

This question is motivated by an obvious formal analogy between two well-known inequalities: Log-concavity and Brunn-Minkowski inequality Let $\mu(dx) := m(x) dx$ be an absolutely continuous ...
12
votes
1answer
321 views

Applications of non-separable Hilbert spaces

In applications, Hilbert spaces of interest are often assumed to be separable. In addition to being extremely convenient mathematically, this assumption can often be justified on computational or ...
2
votes
1answer
165 views

Properties of the weak-$*$ topology

Let $X$ be a topological affine space over a complete base field $\mathbb S := \mathbb C$, $\mathbb R$ or $\mathbb Q_p$. Let $X^*$ be the dual space of continuous affine functionals equipped with the ...
7
votes
7answers
915 views

Gelfand representation and functional calculus applications beyond Functional Analysis

I think it is fair to say that the fields of Operator Algebras, Operator Theory, and Banach Algebras rely on Gelfand representation and functional calculus in a crucial way. I am curious about ...
6
votes
3answers
744 views

problems from the scottish book

Which of the problems from the Scottish Book (pdf of English version) by Stefan Banach are still open? I know that one of the problems was solved by Per Enflo for which he got a live goose from ...
5
votes
2answers
309 views

Do (Banach) ultrapowers carry some sort of 'elementary equivalence'?

The (model-theoretic) ultrapowers had been used for studying elementary equivalnce of first-order structures. Then, they have been adapted to Banach spaces, which are, let me say, second-order ...
32
votes
15answers
6k views

Is there a nice application of category theory to functional/complex/harmonic analysis?

[Title changed, and wording of question tweaked, by YC, because the original title asked a question which seems different from the one people want to answer.] I've read looked at the examples in ...
4
votes
2answers
1k views

Applications of minmax theorem(s)

Intro We suppose $X$ and $Y$ are nonempty sets and f: $X\times Y \rightarrow \mathbb{R}$. A minimax theorem is a theorem that asserts that, under certain conditions, $$ \inf_Y \sup_X f = \sup_X ...
61
votes
15answers
5k views

Why do we care about L^p spaces besides p = 1, p = 2, and p = infinity?

I was helping a student study for a functional analysis exam and the question came up as to when, in practice, one needs to consider the Banach space $L^p$ for some value of $p$ other than the obvious ...
7
votes
3answers
898 views

What are some interesting sequences of functions for thinking about types of convergence?

I'm thinking about the basic types of convergence for sequences of functions: convergence in measure, almost uniform convergence, convergence in Lp and point wise almost everywhere convergence. I'm ...
2
votes
2answers
249 views

Bibliography for topologies defined by a family of seminorms

Hello I am trying to learn more about Fr├ęchet spaces (in order to study the theory of distributions) and was wondering what people thought was the best resource. Thank you very much.