All Questions
13,927 questions
12
votes
1
answer
624
views
Stone–Čech compactification as a semigroup
Let $G$ be a topological group (we can assume that $G$ is countable and discrete) and let $\beta(G)$ be the Stone–Čech compactification of $G$. It is known that $\beta(G)$ can be turned into a left ...
12
votes
2
answers
681
views
Non-sequential spaces in the wild
TLDR: What are examples of (function-)spaces that are not sequential? When does this matter?
As a simple analyst, I am most happy if I can just work with sequences all the time. In most situations ...
- 779
12
votes
1
answer
1k
views
Is there a physical reason that fields in QFT are globally defined?
I have been trying to read a physics textbook on Quantum Field theory. There seems to me to be a bit of a disconnect in most texts I have looked at between quantum mechanics and quantum field theory, ...
- 7,962
12
votes
1
answer
1k
views
Riesz–Markov–Kakutani representation theorem for compact non-Hausdorff spaces
Let $X$ be a compact Hausdorff topological space, and $\mathcal C^0 (X) = \{f:X\to\mathbb{R}; \ f \text{ is continuous }\}$. It is well known that for any bounded linear functional $\phi: \mathcal C^...
- 333
12
votes
2
answers
2k
views
How to think about dual space of a certain space of Lipschitz functions
Consider the following Banach space (for concreteness):
$$X=Lip(\bar{\mathbb{B}}^n)=\{f\in C^0(\bar{\mathbb{B}}^n): \Vert f \Vert_L<\infty \}$$
where
$$
\bar{\mathbb{B}}^n=\{\mathbf{x}\in \mathbb{...
- 2,478
12
votes
3
answers
440
views
Is a certain subset of the disc a convex set?
Some one asked me this question and I thought about it and I don't have any good idea to solve that. Can some one help me and give me an idea to start solve that?
Draw a Cantor set $C$ on the circle ...
- 221
12
votes
2
answers
2k
views
Reference for invariance of essential spectrum under relatively compact perturbations
I'm looking for a proof of the following statement:
Let $X$ be a Banach space and $T$ be a closed map on $X$. For any relatively compact map $A$ the essential spectrum of $T$ and $T+A$ are the same.
...
- 191
12
votes
1
answer
838
views
A measure theory question
Here's an interesting problem one can formulate for a student. This problem arises when considering special ergodic theorems:
On a finite dimensional manifold $M$ with a Lebesgue measure $\mu$, does ...
- 1,143
12
votes
3
answers
530
views
Making an l_2 distance out of l_1 distance
If we think of the l1 distance as a grid-distance between points, then we can think of l2 distance as what we get when we "shortcut" the grid by going "inside" a cell.
Making the grid finer doesn't ...
- 4,462
12
votes
1
answer
353
views
smooth Luzin theorem
For measurable functions $f(x)$, $g(x)$ on $[0,1]$ define the distance $\rho(f,g)$ as a Lebesgue measure of the set $\{x:f(x)\ne g(x)\}$. Then Luzin's famous theorem states that $C[0,1]$ is dense with ...
- 109k
12
votes
1
answer
393
views
Can a non-commutative C*-algebra be a minimal operator space?
By an operator space structure on a Banach space $X$ I mean a sequence of norms on spaces $M_n \otimes X$ that satisfies Ruan's axioms.
Among such admissible sequences there is always the smallest ...
- 5,005
12
votes
2
answers
547
views
Balls in spaces of operators
I am interested in some geometrical aspects of spaces $L(E)$, of bounded operators on a given Banach space $E$. I am unable to estimate if my problem deserves to be asked at MO, but let me try.
Is ...
- 121
12
votes
3
answers
3k
views
Infinitesimal generators of stochastic processes
What's the $L^1$ analogue of Stone's theorem saying that any strongly continuous 1-parameter unitary groups has a unique self-adjoint generator?
More precisely: let $X$ be a measure space ($\sigma$-...
- 22.3k
12
votes
1
answer
510
views
Construction of the universal covering space via compact-open topology
This is a re-post of a question I asked a month ago on MSE, but unfortunately didn't receive any answers. I'm hoping someone could help me with it. Here it goes:
Recently I've been self-studying the ...
- 285
12
votes
1
answer
832
views
Space with semi-locally simply connected open subsets
A topological space $X$ is semi-locally simply connected if, for any $x\in X$, there exists an open neighbourhood $U$ of $x$ such that any loop in $U$ is homotopically equivalent to a constant one in $...
- 303
12
votes
1
answer
645
views
Universal maps between topological spaces
Let $X,Y$ be topological spaces. We call a continuous map $u:X\to Y$ universal if for every continous map $f:X\to Y$ there is $x\in X$ such that $f(x) = u(x)$.
If $u:X\to Y$ and $v:Y\to Z$ are ...
- 48.9k
12
votes
1
answer
885
views
bornological vector spaces over a non-archimedean field
Let $k$ be a complete non-archimedean field. In definitions I have seen of bornological vector spaces over $k$ there are usually some extra assumptions on the non-archimedean field. For instance in '...
- 1,180
12
votes
1
answer
908
views
Equivalence of σ-convex hull and closed convex hull
Let $X$ be a locally convex topological space, and let $K \subset X$ be a compact set. Recalling that the standard convex hull is defined as
$$\text{co}(K) = \Big\{ \sum_{i=1}^n a_i x_i : a_i \geq 0,\,...
- 123
12
votes
2
answers
811
views
Nuclear operators/spaces and transfer operators
While studying for my thesis (in dynamical systems) I've encountered multiple times with the concept of nuclear operators and nuclear spaces, often linked with the works of Grothendieck. For example, ...
- 293
12
votes
1
answer
447
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 ...
- 11.3k
12
votes
1
answer
1k
views
Uniform boundedness of an $L^2[0,1]$-ONB in $C[0,1]$
Assume that we have an orthonormal basis of smooth functions in $L^2[0,1]$. Are there useful practical criteria to determine whether the sup-norm of the basis functions has a uniform bound? I am sure ...
- 4,729
12
votes
1
answer
1k
views
Fixed point theorems and equiangular lines
I've been thinking about the equiangular lines (or SIC-POVM) conjecture, and my conclusion is that the best means of attack would be through some kind of fixed point theorem -- I'm thinking ...
- 6,342
12
votes
2
answers
806
views
Term for a metric space for which the triangle inequality is strict?
Is there a standard term for a metric space in which $\rho(p,r) < \rho(p,q) + \rho(q,r)$ for any distinct $p$, $q$, $r$? Sort of the opposite of metric convexity.
For instance, a subset of ...
- 42.8k
12
votes
2
answers
1k
views
Concrete examples of covering from the 3-torus to the 3-sphere
There is a two-fold branched covering from 2-torus to the 2-sphere, $T^2 \rightarrow S^2$, whose covering transformation group is generated by the map $x \mapsto -x$ (Note that $T^2$ is an abelian ...
- 441
12
votes
2
answers
479
views
C*-algebras with no nontrivial endomorphisms
Pick a C*-algebra $A$ and call a (*-)endomorphism $\alpha:A\to A$ nontrivial if it is injective and $\alpha(A)\neq A$.
Question: Do there exist infinite dimensional C*-algebras with no nontrivial ...
- 1,411
12
votes
1
answer
1k
views
Smoothness of distance function to a compact set
Fix a non-empty compact subset $K\subseteq \mathbb{R}^n$ and let $d_K(x):=\min_{z \in K} \,\|z-x\|$ be the map sending any $x\in \mathbb{R}^n$ to its distance from $K$.
Suppose that:
$K$ is regular : ...
- 5,405
12
votes
1
answer
575
views
Is $\ell_p$ $(1<p<\infty)$ finitely isometrically distortable?
Let $Y$ be a Banach space isomorphic to $\ell_p$, $1<p<\infty$. Is it true that any finite subset of $\ell_p$ is isometric to some finite subset of $Y$?
It seems to me that it is an interesting ...
- 4,878
12
votes
1
answer
727
views
Schemes over topological rings
I have recently been interested in studying an extension of 'usual' algebraic geometry to take into account the topology of $R$ in the definition of the affine scheme $\mathrm{Spec}\, (R)$ when the ...
- 1,270
12
votes
2
answers
878
views
The ground state is signed and symmetric
Background
In Berestycki and Lions it is asserted that (on page 316), if I am not misreading, that the "ground state", i.e. action minimizer among nontrivial solutions, corresponding to the action
$$...
- 39.1k
12
votes
1
answer
329
views
Ideals in smooth subalgebras of C*-algebras
Let $B$ be a $C^{*}$-algebra and $\mathcal{B}$ a dense *-subalgebra stable under holomorphic functional calculus and $C^{1}$-functional calculus for selfadjoint elements. Also, $\mathcal{B}$ is a ...
- 213
12
votes
1
answer
316
views
A reference to a theorem on the equivalence of ideals of measure zero in the Cantor cube
I am looking for a reference of the following (true) fact:
Theorem. For any two continuous strictly positive Borel probability measures $\mu,\lambda$ on the Cantor cube $2^\omega$ there exists a ...
- 42k
12
votes
2
answers
2k
views
Reference on Minty's trick
I am searching for a precise reference for the following result:
Consider $f:\mathbb{R}_+\rightarrow\mathbb{R}_+$ a nondecreasing function.
Assume that a sequence of nonnegative functions $(u_n)_n$ ...
- 3,425
12
votes
1
answer
859
views
Who first found this characterization of Lebesgue integration?
Write $L^1$ for the Banach space $L^1([0, 1])$. Given $f \in L^1$, define $f_1, f_2 \in L^1$ by
$$
f_1(x) = f(x/2),
\qquad
f_2(x) = f((x + 1)/2).
$$
Let $I = \int_0^1$. Then $I$ is the unique ...
- 27.7k
12
votes
1
answer
1k
views
List of all known Riesz representation theorems
Due to the history and development of measure and integration theory and different mathematical schools, there is a huge variety and inconsistency of definitions for concepts like tightness of a ...
- 1,773
12
votes
1
answer
582
views
Is a locally finite union of $G_\delta$-sets a $G_\delta$-set?
Problem. Let $\mathcal F$ be a locally finite (or even discrete) family of (closed) $G_\delta$-sets in a topological space $X$. Is the union $\cup\mathcal F$ a $G_\delta$-set in $X$?
Remark. The ...
- 42k
12
votes
1
answer
217
views
A variant of $\ell^2$-cochains
Suppose $X$ is an infinite countable CW complex which satisfies the following property: for all $k$-cells $e$, the number of $(k+1)$-cells incident to $e$ is at most $c_k$, where the latter is some ...
- 18.8k
12
votes
1
answer
927
views
On an Inequality of Lars Hörmander
Let $P(z)$ be a non-null complex polynomial in $\nu$ variables $z=(z_1,\dots,z_n)$ of degree $\mu$:
\begin{equation}
P(z)=\sum_{|\alpha| \leq \mu} c_{\alpha} z^{\alpha},
\end{equation}
where as usual ...
- 642
12
votes
1
answer
494
views
Does hypoellipticity imply the existence of a parametrix?
Let $M$ be a smooth manifold, like $\mathbb{R}^n$ for instance. The existence of a parametrix for an operator $P$ on $C^\infty(M)$ in any reasonable pseudodifferential calculus implies that $P$ is ...
- 853
12
votes
2
answers
2k
views
Topological Rings
Is it true that, if S is a subring of a separable topological Noetherian ring R,
then S is separable, too ?
- 4,060
12
votes
1
answer
746
views
Which topological spaces are coset spaces of locally compact groups?
What is a topological characterization of the class of spaces that have the form $G/H$ for a locally compact, Hausdorff group $G$ and a closed subgroup $H$ ?
Such a space $X=G/H$ necessarily ...
- 3,497
12
votes
1
answer
1k
views
(Closures of sets of) operations in topological groups.
Let $G$ be a topological group. For each $n \in \mathbb{Z}$, consider the continuous functions $f_{n} \colon G \to G : x \mapsto x^{n}$, and set $F := \{f_{n} \mid n \in \mathbb{Z}\}$.
Is there a ...
- 1,498
12
votes
1
answer
735
views
Parametrisations for null temperature functions: nonuniqueness of solutions to the heat equation
Disclaimer. I expect this is a highly open problem, but maybe I'm wrong and someone has come up with some answers besides those given here. In any case, all information appreciated, thanks!
Definition....
- 1,990
12
votes
1
answer
559
views
Square lying on moving chord of a simple closed curve
Consider a simple closed curve $C$ in $\mathbb{R}^2$. For any points $a$ and $b$ on this curve we associate a point $c_1$ on the left and $c_2$ on the right side to the chord ab, such that $ac_1bc_2$ ...
- 415
12
votes
1
answer
191
views
Spectra on different spaces
This is a method request: I am looking for techniques that allow me to investigate problems like this:
Let $T_1: \ell^1 \rightarrow \ell^1$ be a bounded operator with $\Re(\sigma(T_1)) \subset (-\...
- 305
12
votes
1
answer
777
views
Is every Polish ring topology on $\mathbb{C}$ defined by an absolute value?
There is a unique up to isomorphism algebraically closed field of characteristic 0 and cardinality of the continuum. Let's call it $K$.
We usually call it $\mathbb{C}$, but by this we impose a ...
- 11.6k
12
votes
0
answers
224
views
Do compact inverse-property loops (or just compact Moufang loops) have invariant uniformities and bi-invariant Haar measure?
So, the overall question is in the title: Does a compact topological loop with the inverse property have a Haar measure that is simultaneously left and right invariant? (And we can restrict to ...
- 2,585
12
votes
0
answers
196
views
UMD constant of finite dimensional spaces
For a Banach space $B$, its one-sided Unconditional Martingale Difference (UMD) constant $C^-_p$ (for $p \in (1,\infty)$) is the smallest value such that for all $B$-valued martingale difference ...
- 408
12
votes
0
answers
350
views
Metric completion of an algebraically closed field is algebraically closed?
Let $F$ be a complete metric topological field. Suppose there is a subfield $F_1 \subset F$, algebraically closed and topoolgically dense in $F$. Must $F$ itself be algebraically closed?
We can ...
- 41.1k
12
votes
0
answers
241
views
Is there a characterization of the class of first-order formulas that are closed in every compact Hausdorff structure?
Fix a relational language $\mathcal{L}$. (I don't think relational really matters that much but I don't want to worry about it.) A topological $\mathcal{L}$-structure is an $\mathcal{L}$-structure $M$ ...
- 13k
12
votes
0
answers
329
views
Topology is to semi-decidability, coarse structures are to what?
There is a folklore correspondence between topology as semi-decidability amongst computer scientists, which is explained in places like:
The monograph Synthetic Topology: of Data Types and Classical ...
- 1,221