# Tagged Questions

**12**

votes

**1**answer

224 views

### Discrete subsets in the topology of pointwise convergence vs. metrisability

While reading Arkhangel'skii's Topological function spaces, I encountered an unexpected application of Martin's Axiom. This is Theorem II.5.20:
Assume $\mathsf{MA}+\neg \mathsf{CH}$. Let $X$ be a ...

**4**

votes

**0**answers

131 views

### Ultracoproducts of C(X)-algebras

Let $X$ be a metrizable compact topological space, let $\mathcal U$ be an ultrafilter, and denote by $X^{\mathcal U}$ the ultracopower of $X$ with respect to $\mathcal U$.
As a C$^*$-algebraist, I ...

**8**

votes

**2**answers

319 views

### Ultracoproducts and Cartesian products

Let $X$ be a metrizable compact topological space, let $\mathcal U$ be an ultrafilter, and denote by $X^{\mathcal U}$ the ultracopower of $X$ with respect to $\mathcal U$.
As a C$^*$-algebraist, I ...

**3**

votes

**3**answers

295 views

### Cardinality of $C^*([0,1])$ [closed]

What is the cardinality of the continuous dual of $C([0,1])$ (the set of continuous functions from $[0,1]\to \mathbb{R}$)?

**9**

votes

**1**answer

188 views

### Without AC, which implications between the different definitions of amenability still hold?

More precisely, I would like to know which implications between the following definitions of amenability of a discrete countable (or even finitely generated) group can be proved to hold with only ZF ...

**3**

votes

**1**answer

158 views

### Is it compatible with ZF to assume that every amenable discrete group is finite?

The question is in the title, amenability being understood as the existence of a left-invariant finitely additive probability measure on the group of interest. The case of countable groups is treated ...

**8**

votes

**1**answer

279 views

### Is $\ell^\infty$ Polishable?

Consider $\ell^\infty$ as a subspace of the Polish space $\mathbb{R}^\omega$. It is easy to check that $\ell^\infty$ is not Polish in the subspace topology, as it is countable union of the compact ...

**3**

votes

**1**answer

122 views

### Cardinality of the set of Boolean subalgbras of the lattice of projections on a Hilbert space.

A simple question I've managed to gey myself quite confused about.
Given a Hilbert space H, what do we know about the cardinality of
(a) the set P(H) of projection operators onto H (equivalently, ...

**18**

votes

**1**answer

474 views

### Kaplansky's conjecture and Martin's axiom

Recall Kaplansky's conjecture which states that every algebra homomorphism from the Banach algebra C(X) (where X is a compact Hausdorff topological space) into any other Banach algebra, is ...

**33**

votes

**0**answers

1k views

### Set-theoretic reformulation of the invariant subspace problem

The invariant subspace problem (ISP) for Hilbert spaces asks whether every bounded linear operator $A$ on $l^2$ (with complex scalars) must have a closed invariant subspace other than $\{0\}$ and ...

**9**

votes

**4**answers

613 views

### Is there a maximum to the amount of disjoint non-measurable subsets of the unit interval with full outer measure?

This question arose a few years back when I was an assistant teacher on a course of basic (Lebesgue) measure theory, but I didn't find an answer or anyone able to solve the problem. The setting of the ...

**13**

votes

**2**answers

391 views

### Spectra of elements of a Banach algebra and the role played by the Hahn-Banach Theorem.

This problem was posed on Math StackExchange some time ago, but it did not garner any solutions there. I think that it is interesting enough to be posed here on Math Overflow, so here it goes.
Let $ ...

**4**

votes

**2**answers

245 views

### Cardinality of Equivalence Relation of Eventually Sublinear Functions

Let $\Bbb{R}^{+}\_{0}$ be the set of non-negative real numbers and $\Bbb{R}^{+}$be the set of positive reals. Let us say that a function $f \colon \Bbb{R}^{+}\_{0} \to \Bbb{R}^{+}\_{0}$ is eventually ...

**4**

votes

**1**answer

218 views

### Density character of $\ell_\infty(\kappa, S)$

Let $S$ be an uncountable set. Consider the subspace $\ell_\infty(\kappa, S)$ of $\ell_\infty(S)$ formed by all functions with support of cardinality at most $\kappa$ (here $\kappa<|S|$). ...

**5**

votes

**3**answers

965 views

### Is a semicontinuous real function Borel measurable?

Let $f(x,u): [0,1]^2 \mapsto \mathbb{R}$ be a continuous
function.
[Q] Is $g(x) = \inf_{u\in [0,1]} f(x,u)$ always Borel measurable?
If not, can one find a counter-example?
Note that, for any $c$,
...

**2**

votes

**1**answer

253 views

### Uniqueness of dimension in Banach spaces

Let $\;\;\; \big\langle V,+,\cdot, \;\; ||\cdot|| \;\; \big\rangle \;\;\;$ be a Banach space over the field $\mathbb{K}$, which is either $\mathbb{R}$ or $\mathbb{C}$.
Suppose there exists a subset ...

**16**

votes

**2**answers

2k views

### Is there an infinite-dimensional Banach space with a compact unit ball?

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 ...

**3**

votes

**0**answers

224 views

### What is the origin of the metrization problem for compact convex sets?

The following is an ``old question in analysis:''
Is it true that every perfectly normal compact convex subset of a locally convex topological vector space is metrizable?
Here perfectly normal means ...

**4**

votes

**2**answers

330 views

### Is every bounded representation of Z unitarisable when all sets are measurable?

For the purpuse of this question, a group is amenable iff there exists a Foelner sequence.
Dixmier unitarisability problem asks whether a (countable discrete) group G is amenable iff every bounded ...

**1**

vote

**2**answers

531 views

### Do separable $C^*$-algebras form a set?

The question is in subject.
Update: See Andreas Thom's answer.

**15**

votes

**1**answer

1k views

### Does Arzelà-Ascoli require choice?

Inspired by a recent Math.SE question entitled Where do we need the axiom of choice in Riemannian geometry?, I was thinking of the Arzelà--Ascoli theorem. Let's state a very simple version:
...

**11**

votes

**3**answers

882 views

### Drawing conclusions by NOT using AC.

The existence of non-measurable subsets and functions on $\mathbb{R}$ require the use of the axiom of choice. That is, there exist models of ZF in which all subsets of (and hence all functions defined ...

**11**

votes

**6**answers

1k views

### Finding questions between functional analysis and set theory

Are there some good questions on functional analysis whose solution depends on tools in set theory? My major is mathematical logic, I think tools in set theory, especially infinity combinatorics and ...

**7**

votes

**2**answers

836 views

### Definable collections of non measurable sets of reals

Is there a definable (in Zermelo Fraenkel set theory with choice) collection of non measurable sets of reals of size continuum? More verbosely: Is there a class A = {x: \phi(x)} such that ZFC proves ...

**12**

votes

**1**answer

652 views

### Is Dependent Choice all we really need?

http://en.wikipedia.org/wiki/Axiom_of_dependent_choice
Is DC sufficient for the understanding of objects that are countable in some suitable sense?
For example, is DC sufficient for the full ...

**12**

votes

**2**answers

2k views

### Basis of l^infinity

Is it possible to exhibit a (Hamel) basis for the vector space l^infinity, given by the bounded sequences of real numbers?