All Questions
4,447 questions with no upvoted or accepted answers
3
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114
views
Hereditarily Lindelöf spaces with density continuum
Since there are L-spaces (provably in ZFC), under CH we have regular, hereditarily Lindelöf spaces with density continuum. However, I cannot find an example of such a space under not CH, nor a proof ...
- 31
3
votes
0
answers
198
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On a paper of von Neumann
Let $H$ be a Hilbert space and $T: H \to H$ be a contraction. In Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes, von Neumann proved the inequality
$$
\lVert p(T)\rVert \leq \sup \...
- 31
3
votes
0
answers
161
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Making the powerset into a topological monoid
Every monoid $X$ induces a monoid structure $\circledast$ on $\mathcal{P}(X)$ via
$$U\circledast V := \{uv\ |\ u\in U,v\in V\}.$$
Moreover, a morphism of monoids $f\colon X\to Y$ induces a morphism of ...
- 11.8k
3
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191
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Does "Invariance of domain" hold true for injective Darboux function (instead of continuous injection)?
Let $f \colon U\subset \mathbb{R^n}\to\mathbb{R}^n$ be an injective Darboux map.
Does this imply that $f$ is an open map?
If $f$ is continuous then the result follows from "Invariance of domain&...
- 307
3
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0
answers
86
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(When) can you embed a closed map with finite discrete fibers into a (branched) cover?
Assume all spaces are topological manifolds. A branched cover is a continuous open map with discrete fibers. A finite branched cover is one with finite fibers.
Questions. Given closed map $X\to S$ ...
- 10.5k
3
votes
0
answers
111
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Complex interpolation of Fourier-Lebesgue spaces and Lebesgue spaces
The complex interpolation of Lebesgue spaces $L^{p_0}$ and $L^{p_1}$ are well-known, that is,
$[L^{p_0}, L^{p_1}]_\theta = L^p$, where $(1-\theta)/p_0+1/p_1 = 1/p$ for some $\theta \in [0,1]$.
Now I ...
- 193
3
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141
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Which cardinal $\kappa\geq \omega_1$ is critical for the following property...?
Which cardinal $\kappa\geq \omega_1$ is critical for the following property:
Let $X\subset \mathbb R$ and $\kappa>|X|\geq \omega_1$. Then there is an uncountable family $\{X_{\alpha}\}$ such that $...
- 1,101
3
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0
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295
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Dunford-Pettis like properties for Banach spaces of operators
Let $E$ be a Banach space and $A\subseteq B(E)$ be a Banach subspace of operators on $E$.
Suppose $A$ satisfies the property (RCC) given below:
$$
\left.\begin{array}{l}
(x_n)\subseteq A \textrm{ ...
- 2,605
3
votes
0
answers
115
views
Recovering the matrix when the Schur decomposition of its blocks are known
Let E be a real symmetric matrix in $M_n(\mathbb{R})$ where $ n=2m$ and
$$E=\left(\begin{array}{cc}
G & X \\
X^t & H
\end{array}\right)$$
where $G,H,X$ are $m\times m$ matrices.
Suppose that $...
- 4,058
3
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0
answers
167
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What is the name of the class of topological spaces with the following property ....?
What is the name of the class of topological spaces with the following property $P$ ?
$X\in P$ iff for any open set $W$ in $X$ and any point $x\in \overline{W}\setminus W$ there is an open set $V$ ...
- 1,101
3
votes
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answers
162
views
The essential norm where some Fourier coefficients are fixed
Let us denote $C_{2\pi}$ by the set of all $2\pi$-periodic continuous functions $f:\mathbb{R}\to \mathbb{R}$.
Q. Let $\phi\in C_{2\pi}$. Is the following statement valid?
$$\|\phi\|_2=\inf_{g\in C_{2\...
- 4,058
3
votes
0
answers
88
views
Using a maximum principle to deduce regularity
Suppose $\Omega \subset \mathbb{R}$ is an bounded domain and that $u \in C(0,T; H^{2}) \cap L^{2}(0,T; H^{3})$ where $T >0$.
Consider the PDE on $\Omega \times [0,T]$
$$ \partial_{t}u = a_{1}(x,t) \...
- 131
3
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143
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Extrapolated Integral operator (compactness)
I am studying the compactness of some convolution operators. Let the convolution with extrapolation
$$ \Gamma: X\longrightarrow X; x\mapsto\int_0^t T_{-1}(t-s)B(s)x\mathrm{d}s. $$
Here $T(\cdot)$ is a ...
- 299
3
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209
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Interpolation between Sobolev spaces
In the classical book $Interpolation$ $Spaces$ by Joran Bergh and Jorgen Lofstrom, the Sobolev norm is defined by
$$\|f\|_{H_p^s}=\|D^sf\|_{L^p}$$
where $D^sf$ is defined by the Fourier transform
$$(D^...
- 71
3
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0
answers
82
views
Making a space UMD via interpolation
Recall that a Banach space $B$ has Unconditional Martingale Difference (UMD-$p$) if there is a constant $C_p$ such that for every $B$-valued martingale difference sequences $(d_n)_n$ and choice of $\...
- 408
3
votes
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answers
157
views
Closure of the inverse image under the projection map
Let $S$ be a subsemigroup of a semitopological semigroup $(T,+)$, let $e$ be an idempotent in $T\setminus S$ such that $e\in cl_T(S)$, let $\mathcal{E}$ be a subsemigroup of $S\times S$ such that $(e,...
- 85
3
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109
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"Practical" references on mapping spaces as infinite-dimensional manifolds
I am studying spaces of the form $C^{k}(\mathcal{M},\mathcal{N})$ between manifolds ($k=\infty$ allowed) and I am looking for extensive references, especially analysing their topology and smooth ...
- 1,171
3
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0
answers
123
views
Domain of operator which is used in operator monotone function
We are studying the paper
Rupert L. Frank, Leander Geisinger, Refined semiclassical asymptotics for fractional powers of the Laplace operator, Journal für die reine und angewandte Mathematik (Crelles ...
- 561
3
votes
0
answers
59
views
Sufficient conditions for the weight function to have compact embedding of a weighted Sobolev space
Let $\rho$ be a smooth density function on $\mathbb{R}^N$, that is, $\rho(x)\ge 0$ for all $x\in\mathbb{R}^N$ and $\int_{\mathbb{R}^N}\rho(x) dx=1$. Let $L^p_\rho(\mathbb{R}^N)=\{f: \int_{\mathbb{R}^N}...
- 407
3
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0
answers
175
views
Any reference on Jensen inequality for measurable convex functions on a Hausdorff space?
I asked this question on math.stackexchange and I was suggested that asking it may be more appropriate. This is part of my research which tries to extend some of Choquet's theory to some non-compact ...
- 204
3
votes
0
answers
227
views
How rigorously can we apply the data supplied by this nonstandard attack on Kuratowski's closure-complement problem?
Suppose a student assigned an advanced version of Kuratowski’s closure-complement problem to solve—one that leaves out the standard hint about the finite upper bound of $14$—decides to look for the ...
- 870
3
votes
0
answers
390
views
How to prove the polar decomposition of unbounded operators?
Let $ T $ be a closed, densely defined operator on a Hilbert space $ H $. Then there exists a positive self-adjoint operator $ A $, $ D(A)=D(T) $ and a isometric operator $ V:R(A)\to \overline{R(T)} $ ...
- 1,219
3
votes
0
answers
183
views
Rate of uniform approximation by piecewise constant functions
Definitions and Notation:
Fix a positive constant $M>0$ with positive integers $m,n$ and the standard orthonormal basis $e_1,\dots,e_n$ of $\mathbb{R}^n$.
For every positive integer $N$, define the ...
- 5,405
3
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161
views
Ergodic diffeomorphisms of the circle
From the paper
Halmos, Paul R., In general a measure preserving transformation is mixing, Ann. Math. (2) 45, 786-792 (1944). ZBL0063.01889.
the following result is known: Let $(E,\Sigma, \mu)$ be a ...
- 101
3
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110
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On the relation between ellipticity and Fredholmness as properties of linear PDE's on Fréchet spaces of smooth sections
Let $M$ be a compact manifold equipped with finite rank vector bundles $E$ and $F$ with spaces of $C^{\infty}$ sections denoted $\Gamma(E)$ and $\Gamma(F)$ respectively. It is standard that a ...
3
votes
0
answers
116
views
Malliavin-Shavgulidze (type) measures on the group of measure-preserving invertible maps on $\mathbb T$?
The Malliavin-Shavgulidze measures on $\operatorname{Diff}^{1}(I)$ (with $I$ an interval of $\mathbb R$) are defined as the image $W_{\sigma} \circ f^{-1}$ of the Wiener measure $W_{\sigma}$ with ...
- 101
3
votes
0
answers
117
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Arithmetic progressions and removal lemmas for graphs in arithmetic combinatorics
As it is well known, one can gets a proof of Roth's Theorem concerning arithmetic progressions of length 3 (APs for short) by using the celebrated Ruzsa-Szemerédi triangle removal lemma for graphs.
In ...
- 1,561
3
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106
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Two topologies on the space of maps from an algebraically closed field to a projective variety
This question is related to this one but I have written this in a self-contained manner.
All varieties are complex varieties.
For quasi-projective variety $U$ and a projective variety $X$ we can ...
- 5,901
3
votes
0
answers
88
views
Is the thickening of a PL 2-disc in $\Bbb R^4$ a 4-ball?
Let $D\subset\Bbb R^4$ be a PL-embedded 2-dimensional disc. Let $N=D+K$ be a thickening of the disc, where $K$ is some sufficiently small 4-dimensional PL-ball and "$+$" means Minkowski ...
- 13.6k
3
votes
0
answers
31
views
Compactness of the minimal ideal of a compact Hausdorff polytopological semigroup
A semigroup $X$ endowed with a topology is called
$\bullet$ a topological semigroup if the semigroup operation $X\times X\to X$ is continuous;
$\bullet$ a semitopological semigroup if for every $a,b\...
- 42k
3
votes
0
answers
212
views
Two equivalent definitions of semisimplicity of group representations, proof by Zorn's lemma, a “counterexample” from the Fourier transform theory
Consider a representation $A$ of a group $G$ in a complex vector space ${\mathbb{V}}$:
$$
A:~~G~\longrightarrow~\operatorname{GL}({\mathbb{V}})~~,
$$
and let ${\mathbb{V}}$ be decomposable into a ...
- 603
3
votes
0
answers
75
views
Discreteness of the higher inductive-inductive Cauchy real numbers in real cohesive homotopy type theory
We work in cohesive homotopy type theory with propositional resizing, so that there is only one type of Dedekind real numbers $\mathbb{R}$ up to equivalence, and Mike Shulman's axiom $\mathbb{R}\flat$,...
- 2,624
3
votes
0
answers
105
views
Space of algebraic maps and quotient under finite group action
For a normal (you can assume smooth for this problem) quasi projective complex variety $X$ and a projective complex variety $Y$, we can endow the space of the set of morphisms $\operatorname{Mor}(X,Y)$...
- 5,901
3
votes
0
answers
99
views
Definition clarification: "regular directed distributions"
(I asked this question on math.stackexchange (see here) but didn't receive any reaction, hence I try it here. If it does not fit within here, just let me know in the comments.)
In the definition of ...
- 1,171
3
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0
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447
views
Equivalence of weak and weak-* convergence for sequences and reflexivity
Let $X$ be a Banach space and $X^*$ its topological dual space. Let us define the property (WS):
For all sequences $(x_n^*) \subset X^*$ and all $x^* \in X^*$, we have
$$x_n^* \rightharpoonup x^* \...
- 1,724
3
votes
0
answers
126
views
A path with zero increments and positive area
I am studying rough paths from the 2007 St Flour lecture notes and I came across the example at the end of chapter one of the sequence of paths $X(n):[0,2\pi]\to \mathbb R^2$ given by $X_t(n) = \frac{...
- 177
3
votes
0
answers
204
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Beurling's theorem on invariant subspaces
Beurling's theorem characterize the closed subspaces $M\subset H^2$ of the Hardy space, which are invariant under the shift operator $Sf(z):=zf(z)$, as spaces of the form $\varphi H^2 $ where $\varphi$...
3
votes
0
answers
153
views
Is it possible to reconstruct the universally measurable sets in X from the $C^*$-algebra $C(X)^{**}$?
This continues my question of two months ago. Let $X$ be a compact Hausdorff topological space. We consider the $C^*$-algebra $C(X)$ of continuous functions on $X$, its dual space $C(X)^{*}=M(X)$ of ...
- 7,426
3
votes
0
answers
81
views
Example of the bounded convolution operator when Sharpley's conditions does not hold
I am reading about Orlicz and Marcinkevich spaces, and wondering whether there is an example in which Sharpley's condition is not satisfied for a special bounded operator $T_k$ (see for reference ...
- 97
3
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0
answers
135
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Holmgren's theorem on the boundary
Consider $\Omega$ a bounded Lipschitz domain, with $\gamma \subseteq \partial \Omega$ a $C^2$ manifold. I am interested in proving the following.
Let $u: \Omega\times [0,T]\rightarrow \mathbb{R}$ be ...
- 235
3
votes
0
answers
79
views
Is $\mathfrak q_0$ equal to the smallest cardinality of a second-countable $T_1$-space which is not a $Q$-space?
A topological space $X$ is a $Q$-space if every subset of $X$ is of type $G_\delta$.
The smallest cardinality of a metrizable separable space which is not a $Q$-space is denoted by $\mathfrak q_0$ and ...
- 42k
3
votes
0
answers
130
views
Is the range of the exterior covariant derivative closed in $L^{2}$?
Let $(M,g)$ be a compact Riemannian manifold. Given a tensor bundle $\mathbb{E}$, let $\nabla:\Gamma(\mathbb{E}) \rightarrow \Gamma(T^{*}M\otimes \mathbb{E})$ be the canonical connection induced by ...
- 808
3
votes
0
answers
174
views
On continuous seminorms on Fréchet-Stein algebras
Let $K$ be a discretely valued complete non-archimedean field and $U$ be a left Fréchet-Stein algebra as defined in Algebras of p-adic distributions and admissible representations, with a Fréchet-...
- 541
3
votes
0
answers
219
views
Can any POVM be induced by a quantum instrument?
I suspect this is the obvious result of something in operator algebras, but that's far outside my field.
Recall that a projection-valued measure is a map $E$ from a sigma-algebra $\mathcal{F}$ on ...
- 854
3
votes
0
answers
110
views
Tauberian theorem with flatness condition
Suppose $f(z)=\sum_{n=0}^{\infty}a_nz^n$ is a series with $a_n\in \mathbb{R}$ and radius of convergence $1$ and such that $f$ restricted to $[0,1[$ admits a smooth extension to $[0,1]$ with $f^{(n)}(1)...
- 278
3
votes
0
answers
257
views
Complex Hölder space
I already posted this question on math.stackexchange, but got no response and was suggested to post it here.
I came across a space in an ergodic theory paper, which I am calling here a (complex) ...
- 136
3
votes
0
answers
430
views
What's the problem with the evaluation map not being continuous?
When introducing differentiable functions between locally convex spaces, many authors (e.g. Bastiani, Keller, Kriegl-Michor) notice that the evaluation map
$$ E\times E^*\to\mathbb R,\qquad (x,L)\...
3
votes
0
answers
43
views
Continuous analogue for Szpilrajn Theorem: complete preorder extends a continuous preorder
A corollary of Szpilrahn Theorem states:
Any preorder on nonempty $X$ has a complete and transitive extension.
I am thinking about the "Szpilrahn Theorem" for continuous preorder on ...
- 599
3
votes
0
answers
190
views
$C^1$-regularity of solution of a Dirichlet problem
I've been stuck trying to hunt down a proof/reference for a certain regularity result which I need for my thesis (all the authors I've consulted refer to it as "well-known"). Consider the ...
3
votes
0
answers
168
views
Besov space norms
We need to recall some Besov space norms to formulate the question.
Let $d \in \mathbb N$, $0<s<2, 1 \le p,q \le \infty$. Then the Besov space $B^s_{p,q}(\mathbb R^d)$ is given by the norm
$$ \...
- 475