All Questions
13,927 questions
3
votes
1
answer
263
views
Give an example of a star-Menger space which is not star-$K$-Menger
A space $X$ is said to be star-Menger if for every sequence $(\mathcal{U}_n)$ of open covers of $X$ there exists a sequence $(\mathcal{V}_n)$ such that for each $n$ $\mathcal{V}_n$ is a finite subset ...
CommunityBot
- 1
0
votes
0
answers
101
views
Finding an example if it exists, for a non-contractible and contractible space with special requirement on quotients of their union?
Let $A$ and $B$ be subsets of $n$-dimensional Euclidean space $\mathbb{R}^{n}$, such that $A$ is non-contractible, $B$ is contractible and $B$ is not an one-point set.
I would like to find example(s) ...
0
votes
0
answers
51
views
Approximating open subset of profinite group by union of cosets of ideal
I am trying to understand the proof of Theorem 1.3 in this paper by poonen. Poonen refers to Lemma 20 in a different paper. He claims that the open subset $U_P \subseteq \hat{\mathcal{O}}_P$ can be ...
- 1
6
votes
1
answer
284
views
When does $\mathbf{Top}/X$ embedd fully faithfully into $\mathbf{Top}$?
Under what conditions on the topological space $X$ is the overcategory $\mathbf{Top}/X$ of topological spaces over $X$ equivalent to a full subcategory of $\mathbf{Top}$? Surely if $X$ terminal i.e. a ...
CommunityBot
- 1
3
votes
1
answer
621
views
What is the Lebesgue covering dimension of this topological space?
Take the 4 dimensional time-oriented spacetime $(M,g)$ such that it's not strongly causal.
Take the induced topology defined by the Lorentzian metric called Alexandrov topology.
This topology matches ...
- 612
1
vote
0
answers
87
views
$f \in L^2(X\times Y,\mu \times K)$ for Kernel $K$, is the map $X \ni x \mapsto (f(x,\cdot),x) \in \bigsqcup_{x \in X}L^2(Y,\Sigma_Y,K_x)$ measurable?
Let $(X,\Sigma_X)$ and $(Y,\Sigma_Y)$ be two measurable spaces, let $\mu$ be a measure on $(X,\Sigma_X)$, and let $(K_x)_{x \in X}$ be a transition kernel from $(X,\Sigma_X)$ to $(Y,\Sigma_Y)$, that ...
- 309
13
votes
0
answers
573
views
Classical (i.e. commutative) spaces with quantum symmetry but no classical symmetry
In a recent preprint (arXiv:2311.04889), my coauthors and I constructed a sequence of graphs with no classical symmetry which nevertheless have quantum symmetry.
For graphs this had been an open ...
- 2,339
0
votes
0
answers
28
views
Metric entropy of mixed norm spaces with exponent-free bounds
Suppose $\mathcal{F}\subset L^p([0,1]^d)$ is a subset with the following property: The $L^q$-covering number of $\mathcal{F}$ is independent of $q$, for all $1\le q\le\infty$. An example of $\mathcal{...
- 21
3
votes
0
answers
84
views
About the naturality of Krasnoselskii genus on Variational Methods
I have recently watched a seminar about Variational Methods from Mónica Clapp and she gave a very interesting motivation of why the Lusternik–Schnirelmann category (click on the link for the ...
- 131
4
votes
0
answers
135
views
Automorphism-invariant positive linear functionals on $C*$-algebras
Let $A$ be a $C^*$-algebra. Does there exist a non-trivial positive linear functional $\nu\in A^*$ which is $\mathrm{Aut}(A)$-invariant? That is, $\nu\circ\alpha=\nu$ for all $\alpha\in\mathrm{Aut}(A)$...
- 1,959
4
votes
0
answers
148
views
Is every pointwise-weakly continuous one-parameter group of automorphisms of B(H) given by a Hamiltonian?
Let $\mathcal H$ be a Hilbert space, $\mathscr B(\mathcal H)$ be the von Neumann algebra of all bounded operators on $\mathcal H$, and let $\sigma $ be a one-parameter group
of automorphisms of $\...
- 2,263
3
votes
0
answers
105
views
Finitely generated Banach lattice $C(K)$ and partitions of unity
Let $E$ be a Banach lattice. A Banach sublattice $L$ of $E$ is called finitely generated if there exists a finite subset $F \subseteq E$ such that
$$L = \bigcap \{ \hat{L} \mid F \subseteq \hat L, \, \...
- 63.9k
11
votes
1
answer
309
views
Which closed subsets $Y$ of a compact space $X$ admit a linear extensor $C(Y)\to C(X)$?
In the following $X$ is a Hausdorff compact topological space. Let $Y$ be a closed subset of $X$.
The restriction operator $R_Y:C(X)\to C(Y)$ is surjective (Tietze), so it admits a continuous right ...
- 5,596
5
votes
0
answers
94
views
When a compact subset of a TVS can be continuously projected on a closed linear subspace?
Let $V$ be a (Hausdorff) topological vector space, $W\subset V$ a closed linear subspace, $X\subset V $ a compact.
(Q):
When there is a continuous map $P:X\to W$ such that $P(x)=x$ for every $x\in X\...
- 60.6k
6
votes
1
answer
403
views
Do acyclic amenable groups exist?
Is there an example of a nontrivial discrete amenable group with vanishing integral homology?
To put the question in contrapositive. Given arbitrary acyclic group $Q$, is there some reason for the ...
- 136
3
votes
0
answers
132
views
Is the Schwartz space a tame Frechet space?
I ran into the following definition of tame Frechet spaces and Nash-Moser therem.
It says that the space of smooth functions on a compact manifold is tame Frechet.
However, I wonder if
The Schwartz ...
- 3,477
0
votes
1
answer
731
views
Does weak convergence in $L^2$ imply convergence a.e. of a subsequence? [closed]
The title pretty much explains it all. Let $u_n\in L^2(\mathbb{R}^n)$ be a sequence converging weakly in $L^2$ to some $u\in L^2(\mathbb{R}^n)$, that is $\int u_k v \to \int u v$ for all $v\in L^2(\...
- 1,989
8
votes
1
answer
307
views
Is the dual of a Fréchet space weakly* separable?
It is known that if $X$ is a separable Banach space with dual $X^\ast$, then $B_{X^\ast}$, the closed the unit ball in $X^\ast$, is compact and metrizable in the weak* or $\sigma(X^\ast, X)$-...
- 60.6k
1
vote
1
answer
171
views
How is interpolation used in the proof of Lemma 4.1 in Tao's article Endpoint Strichartz Estimates?
In the proof of Lemma 4.1, pp. 962–963 in "Endpoint Strichartz Estimates" by Tao and Keel (1997) (see MR1646048 or Zbl 0922.35028), the authors first proved the statements hold for some ...
- 47.5k
2
votes
1
answer
213
views
Are projective tensor products left-exact if one considers only maps of norm at most 1?
Consider the category $\mathrm{Ban}$ of Banach spaces and bounded linear maps and the category $\mathrm{Ban}_1$ of Banach spaces and bounded linear maps of operator norm at most 1. Let $\otimes_\pi$ ...
- 9,486
0
votes
3
answers
278
views
A generalisation of Tchebychev inequality
Let $f,g \in C(\mathbb R)$ with $\exists M \in \mathbb R^*, \forall (x,y) \in \mathbb R^2, M\times (f(x)-f(y))(g(x)-g(y)) \geq 0$.
Is it true that exists $ u$ any real function, and $a,b$ monotone ...
- 12.9k
1
vote
2
answers
498
views
Show convergence result
Consider the following sets:
$$
A = \Big\{ x\in X: \Pr\bigg(\lim_{n \to \infty}d\big(p_n, [\ell(x), u(x) ] \big)= 0\bigg)=1 \Big\},
$$
and
$$
A_n = \Big\{ x\in X: d\big(p_n, [\ell(x), u(x) ] \big)...
- 108
5
votes
0
answers
899
views
Is such a Banach space $X$ isometrically isomorphic to a Hilbert space or not?
Let $X$ be a real or complex Banach space. $X$ satisfies:There exists real or complex series $\{a_k\}_{k=1}^n,\{b_k\}_{k=1}^n$ (which satisfies that:$\begin{cases}
a_k,b_k\in \mathbb{R}\ \forall k=1,\...
- 181
2
votes
1
answer
205
views
$L^2$ space of Hilbert-Schmidt operator valued functions
Let $\mathscr{S}$ denote the space of all Hilbert-Schmidt operators on $L^2(\mathbb R)$. Consider the Hilbert space $L^2(\mathbb R, \mathscr S)$ of square-integrable $\mathscr S$-valued function, that ...
1
vote
0
answers
56
views
Convergence of slice in an equivalent renorming
Let us consider $\ell_2$ space with $\Vert \cdot \Vert_2$ norm. Let us define a new norm equivalent to $\Vert \cdot \Vert_2$ norm as follows:
$$
\Vert x \Vert_0 = \max \{ \Vert x \Vert_2, \sqrt{2} \...
- 85
24
votes
3
answers
3k
views
The closure-complement-intersection problem
Background
$\DeclareMathOperator\Cl{Cl}$
Let $A$ be a subset of a topological space $X$. An old problem asks, by applying various combinations of closure and complement operations, how many distinct ...
- 63.9k
2
votes
1
answer
272
views
Decompose a function into a bounded part and a Lipschitz part
Let $f: \mathbb R^d \to \mathbb R^d$ be a measurable function such that
$$
\sup_{x,y \in \mathbb R^d} \frac{|f(x) - f(y)|}{\max \{1, |x-y| \}} < \infty.
$$
Are there functions $g,h: \mathbb R^d \...
- 42.8k
2
votes
2
answers
197
views
$L^p$ domination of mixed partial derivatives by the unmixed ones?
Is it true that for each real $p\ge1$ there is some real $C_p$ such that for all smooth real-valued functions $u$ compactly supported on $S:=(0,1)^2$ one has
$$\|D_1D_2u\|_p\le C_p(\|D_1^2u\|_p+\|D_2^...
5
votes
2
answers
1k
views
How can I prove this special version of the Poincaré inequality?
I am reading a paper, Is $L^2$ Physics-Informed Loss Always Suitable for Training Physics-Informed Neural Network?, which uses an "application" of Poincaré's inequality. The form I know and ...
- 128k
2
votes
1
answer
154
views
Are these two norms on localized versions of $L^p_q$ equivalent?
$\newcommand{\RR}{\mathbb R}\newcommand{\diff}{\, \mathrm d}$ We fix $T \in (0, \infty)$ and $p, q \in [1, \infty)$. Let $\mathbb T$ be the interval $[0, T]$.
Let $E$ be the space of all real-valued ...
4
votes
1
answer
279
views
Schroedinger operator in 2 dimensions with singular potential
Consider the Schroedinger operator
$$H = -\Delta + \frac{c}{\vert x \vert^2}$$
in two dimensions with $c >0$
This operator has a self-adjoint realization, since it is a positive symmetric operator ...
- 24.2k
18
votes
3
answers
1k
views
Is there a natural measurable structure on the $\sigma$-algebra of a measurable space?
Let $(X, \Sigma)$ denote a measurable space. Is there a non-trivial $\sigma$-algebra $\Sigma^1$ of subsets of $\Sigma$ so that $(\Sigma, \Sigma^1)$ is also a measurable space?
Here is one natural ...
- 4,713
0
votes
0
answers
52
views
Estimate the gradient (with respect to local coordinates) of a partition of unity on a manifold
Suppose $\{U_\alpha\}$ is an atlas of coordinate patches of a (noncompact) smooth manifold $M$ of dimension $n$, with coordinates $(x_\alpha^1,\dots,x_\alpha^n)$ on $U_\alpha$. Furthermore we assume ...
- 21
2
votes
0
answers
56
views
Fractional powers of Dirichlet-to-Neumann map to derive estimate for PDE
Assume $\Omega$ is an open, bounded subset of $\mathbb R^3$ with smooth boundary $\partial \Omega= \Gamma$. For $u \in
H^{1/2}(\Gamma)$, let $U \in H^1(\Omega)$ denote the weak solution of
the ...
- 93
0
votes
0
answers
220
views
Eigenvalue multiplicity of tensor product of positive operator with itself
Let $H$ be a separable complex Hilbert space and let $A\in B(H)$ be positive with $||A||=1$ and have eigenvalue 1 with multiplicity 1. Suppose $A=T^*T$ for some $T\in B(H)$. Denote the spectrum of $A$ ...
- 31
1
vote
0
answers
94
views
positive invertible maps which are not *-automorphisms
Let $A$ be a unital C*-algebra. Is there a unital positive self-map $F:A\to A$ which is invertible (i.e. injective and surjective) but not a $*$-automorphism? If yes, how does appear its Gelfand-...
- 41
0
votes
1
answer
140
views
Singular integral bounded by Dirichlet form?
We define for some fixed $L$
$$\Omega:=\{(x_1,x_2) \in ([-L,L]^2 \times [-L,L]^2) \setminus \{x_1=x_2\}\},$$
in particular $x_1,x_2 \in \mathbb R^2.$
Let $f \in C_c^{\infty}(\Omega)$, then I am ...
- 128k
1
vote
1
answer
101
views
Image of a complete topological group under open and surjective map is complete?
A uniform space $X$ is complete if every Cauchy filter in $X$ is convergent. Here we do not require $X$ to be Hausdorff.
Question.
Let $G$ be a complete topological group and let $H$ be a topological ...
CommunityBot
- 1
0
votes
0
answers
60
views
asymptotic expansions for $C^{1+\epsilon}$operators
I want to understand the asymptotic expansion for $C^{1+\epsilon}$ operators.
More precisely, if a complex operator $T(z)=\sum_{n \ge 1}T_n z^n$ is defined on a closed unit disk, and it is $C^{1+\...
- 6,367
2
votes
1
answer
173
views
Difference in essential spectrum between Schrodinger operators
I am considering two Schrodinger operators on $\mathbb{Z}^2$ and compare their essential spectrum. The operators are both of the form $H=A+V$ where $A$ is the adjacency operator on the $\mathbb{Z}^2$-...
- 633
46
votes
2
answers
5k
views
Continuous bijections vs. Homeomorphisms
This is motivated by an old question of Henno Brandsma.
Two topological spaces $X$ and $Y$ are said to be bijectively related, if there exist continuous bijections $f:X \to Y$ and $g:Y \to X$. Let´s ...
- 11.5k
4
votes
0
answers
290
views
Which countable sets don't drastically change the definable topologies on $\mathbb{R}$?
For $\mathcal{M}$ an expansion of $\mathcal{R}=(\mathbb{R};+,\times)$ and $A\subseteq\mathbb{R}$, let $\tau^\mathcal{M}_A$ be the topology on $\mathbb{R}$ generated by the sets definable in $\mathcal{...
- 1
3
votes
1
answer
163
views
Rescaling Fourier coefficients of a continuous function by a bounded sequence
This question stems out of: which sequences $(a_n)_{n\in\mathbb{Z}}$ of complex numbers have the property that if there exists a continuous function $f$ on the circle with Fourier coefficients $b_n$, ...
- 24.2k
6
votes
2
answers
290
views
If a Banach / Fréchet manifold $M$ happens to be a topological vector space, is $M$ just a Banach / Fréchet space?
In finite dimensions, if $M$ is a smooth manifold that happens to be a vector space, then it is indeed just the Euclidean space.
I wonder if the same result holds valid in infinite dimensions. More ...
- 3,159
1
vote
1
answer
83
views
Convexity property of an equivalent norm on $\ell_2$
Let us consider the space $\ell_2$ with an equivalent norm defined by
$$
\Vert x \Vert = \max \{ \Vert x^{'} \Vert_2, \Vert x^{''} \Vert_2 \},
$$
where $x^{'}=(0, x_2, x_3, \cdots)$, $x^{''} = (x_1, 0,...
- 128k
4
votes
0
answers
73
views
Find reasonable definition for endpoint Lorentz function spaces $L^{\infty,q}$ via the idea from endpoint Triebel-Lizorkin ${\scr F}_{\infty,q}^s$
On a measure space $(X,\mu)$, for $0<p,q<\infty$ the Lorentz space $L^{p,q}(\mu)$ is defined by $$\|f\|_{L^{p,q}(\mu)}:=p^\frac1q\|t\mu(|f|>t)^\frac1p\|_{L^q(\mathbb R_+,\frac{dt}t)}=p^\...
- 938
4
votes
1
answer
287
views
Reference request: Gaussian measures on duals of nuclear spaces
I am interested in constructive quantum field theory where Gaussian measures on duals of nuclear spaces (specifically, the space of tempered distribution $\mathcal{S}'(\mathbb{R}^n)$) play a key role. ...
1
vote
0
answers
67
views
Is the adjoint action of $\mathrm{SU}(2)$ on the Schwartz space proper and free?
$\DeclareMathOperator\SU{SU}$Let $t_1, t_2, t_3$ be generators of the Lie algebra $\mathfrak{su}(2)$.
Let us consider a Schwartz space $\mathcal{S}$ defined as
\begin{equation}
\mathcal{S}:= \Bigl\{ \...
- 63.9k
2
votes
1
answer
140
views
Invertibility of message passing with invertible parametrization
Consider the message passing framework defined by, $$f(\boldsymbol{x}_i)= \boldsymbol{x}_i + \sum_{j \neq i} (\boldsymbol{x}_i -\boldsymbol{x}_j) g(\|\boldsymbol{x}_i -\boldsymbol{x}_j\|^2),$$ for $i=...
- 128k
1
vote
1
answer
95
views
Maximal element w.r.t. abolute continuity of measures
Suppose that $\mu$ is a $\sigma$-finite measure on $\mathcal{X}\equiv\bigotimes_{i=1}^n\mathcal{X}_i$. Let $\Pi$ denote the set of all $\sigma$-finite product measures on $\mathcal{X}$. Define
$$
\...
- 24.2k