All Questions
4,450 questions with no upvoted or accepted answers
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Is $(\omega+1)^\omega$ with the box topology ultraparacompact?
Let $\omega+1$ be endowed with the interval topology, that is $U\subseteq (\omega+1)$ is open if $U\subseteq\omega$ or $(\omega+1)\setminus U$ is finite. We call $U\subseteq (\omega+1)$ basic if ...
- 48.9k
7
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927
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What's the idea behind various equivalent norms on Besov spaces $B^{s}_{p,q}$?
I am trying to understand Besov spaces; and I am eager to see why the various norms are equivalent on it.
Let $\phi$ be a $C^{\infty}$ function on $\mathbb R^{n}$ with
$ \operatorname{supp} \phi \...
- 1,051
7
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0
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744
views
What function space does holomorphic functional calculus give us?
Let $A$ be a unital Banach algebra, $U$ be an open subset of $\mathbb{C}$, and $A_U:=\{x\in A:\sigma(x)\subset U\}$. Holomorphic functional calculus says that any holomorphic function $f:U\rightarrow\...
- 175
7
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559
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The Banach space of bounded functions with countable support
Let $X$ be a set of cardinality $\aleph_1$ and consider the Banach space $\ell_\infty^c(X)$ of all scalar-valued bounded functions on $X$ which are non-zero only for countably many elements of $X$ ...
- 11.3k
7
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116
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Bundles over Function Spaces
Is there any reference on bundles over function spaces? In particular, I am interested in Banach-bundles over function spaces like $W^{k,r}(M)$, where $M$ is a Riemannian manifold. Separable Hilbert-...
- 71
7
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289
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Trace class norms of special integral operators
Let $\mu$ be a finite compactly supported Borel measure on the real line. On the space $L^2(\mu)$ consider the integral operators
$$
(K_a f)(x)=\int k_a(x, y)f(y)d\mu(y)
$$
with
$$
k_a(x, y)=\frac{a\...
- 452
7
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234
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Is there a tensor norm that preserves Rosenthal Banach spaces?
By a Rosenthal Banach space I mean one that does not contain an isomorphic copy of $\ell_1$. Is there a tensor norm $\alpha$ such that the Banach tensor product $E\otimes_\alpha F$ is Rosenthal if $E$ ...
7
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550
views
Zariski-homeomorphisms
This question is motivated by two questions at MO and
at MSE.
I am interested in homeomorphism types of (irreducible) complex-projective varieties with respect to the Zariski topology. Any two ...
- 12.3k
7
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0
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293
views
Complex interpolation of a Banach space and its antidual when the space has a basis
Given a complex Banach space $X$ and its antidual $\hat X^*$, it is possible in some cases to apply the complex interpolation method, and get as $(X,\hat X^*)_{1/2}$ a Hilbert space. See [F. Watbled, ...
- 4,461
7
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2k
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Prokhorov's theorem for finite signed measures?
Prokhorov theorem provides a useful characterization of relatively compact sets w.r.t. narrow topology (topology induced by narrow convergence) in the space of probability measure.
Notation used ...
- 339
7
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0
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199
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Central Extension of Continuous Loop Group
For the group $LG$ of smooth loops into a simple compact 1-connected Lie group $G$ there is a well-known universal central extension. My qustion is basically whether this extension also exists for the ...
- 1,040
7
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0
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708
views
Minkowski's Inequality for Integrals in Orlicz spaces
EDIT: I have changed the question to have less parameters, fitting it into the context of Orlicz spaces.
Suppose $f:[0,\infty)\to[0,\infty)$ is convex and increasing, $f^{-1}:[0,\infty)\to[0,\infty)$...
- 882
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466
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Closure properties of familes of $G_\delta$ sets.
Given a family of sets $G\subset P(X)$, can one characterize by "closure properties" alone whether or not $G$ arises as the family of all $G_\delta$ for some topology on $X$? some Polish space ...
- 17.6k
7
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299
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Generalized Skorokhod spaces
Skorokhod spaces of càdlàg functions are an extremely useful setting to describe stochastic processes. I'd like to understand the Skorokhod topology from a pure topological point of view, without ...
- 8,512
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355
views
An $L^{\infty}$ version of principal component analysis?
I have a $k$ by $n$ matrix $A$, with $k \ll n$. In case it helps, the $k$ rows are orthonormal.
I'm interested in finding a $k$ by $k$ orthogonal matrix $M$ so as to maximize the $L^{\infty}$ norms ...
- 193
7
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626
views
Does local strict contractibility imply ANR?
Say that a space (= compact metrizable space) $X$ is locally strictly contractible if, for every $p\in X$ and neighborhood $U$ of $p$, there is a neighborhood $V$ of $p$ which can be contracted to $p$ ...
- 2,049
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answers
517
views
Is there a natural topology on the set of open sets ?
Given a topological space $(X,\mathcal{O})$ can one assign a natural topology to $\mathcal{O}$ such that
1) The intersection of a compact set of open sets is again open,
2) The maps $\cap,\cup:\...
- 11.1k
7
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0
answers
266
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Problem with Shelah and Stern's paper on the Hanf number of the theory of Banach spaces
I have been trying to understand "The Hanf number of the first order theory of Banach spaces" by Shelah and Stern (Trans. AMS 244 (1978) 147-241). They construct a normed space $M$ from a Hilbert ...
- 879
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624
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"Liftings" of L^\infty functions
This is motivated by this question: Is there an inclusion of $L_\infty(G)$ into $C_0(G)^{**}$? and Bill Johnson's comments there.
Let $X$ be a locally compact Hausdorff space and $\mu$ a Radon ...
- 18.7k
7
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0
answers
140
views
integral transforms defined by polygons
Are there any literature on operators of the form
\[ (Tf)(x) = \int K(x,y)f(y), dy\]
where $K(x,y)$ is the characteristic function of a polygon in $\mathbb{R}^2$. I would like to know if the spectrum ...
- 22.8k
7
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161
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Seeking reference - criterion for the existence of a positive linear functional on an ordered vector space below a given function
The following surely appears somewhere, I would greatly appreciate a reference. (The aim is to get a measure via Riesz representation, but that has nothing to do with the statement.)
Let $X$ be an ...
- 577
7
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310
views
The self-duality of topological compactness
The impatient reader can skip my attempt at motivation and go straight my "Question formulations for the impatient."
In a failed(?) attempt at discovering something new, some years ago I ...
- 17.6k
7
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1k
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Reference request: Arzela-Ascoli theorem for smooth Hölder norms
Could anyone suggest a textbook account of the Arzela-Ascoli theorem for $C^{k,\alpha}$ norms?
- 29.1k
7
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433
views
Ever seen a ringed group?
A locally ringed space is a common generalization of schemes and various manifolds. I am wondering about a locally ringed group which should be a common generalization of group schemes and various Lie ...
- 12.3k
6
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0
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212
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Hölder's inequality for trace-class maps of $p$-liquid spaces and a related conjecture of Grothendieck
In Condensed Math and Complex Geometry Proposition 8.8, Clausen-Scholze describe trace-class maps between projective objects in the $p$-liquid category as sums of rank 1 operators against ${<}p$-...
- 61
6
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98
views
Rigid plane curves
A curve is a continuous one-to-one image of the real line $\mathbb R$.
A space $X$ is rigid if the only homeomorphism of $X$ onto itself is the identity.
Is there a rigid curve in the plane?
I am ...
- 3,317
6
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162
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Dual space of local Sobolev space on a manifold
$\newcommand{\comp}{\mathrm{comp}}$As part of my master's thesis, I am currently learning about Sobolev spaces on manifolds. From my research online, I found out, that there are a lot of ways to ...
6
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159
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Identification of Fock space and the $L^2$ space of tempered distributions
Let $\mathcal{S}'(\mathbb{R}^d)$ be the set of tempered distributions over $\mathbb{R}^d$ and $d\phi_C$ a Gaussian measure over $\mathcal{S}'(\mathbb{R}^d)$ with covariance operator $C$. Consider the ...
- 721
6
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293
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Looking for Mackey's PhD thesis, "The subspaces of the conjugate of an abstract linear space"
I'm looking for a copy of George Mackey's PhD thesis, The subspaces of the conjugate of an abstract linear space (Harvard Univ., 1942), but am currently struggling to find one online, with the only ...
- 11.8k
6
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answers
197
views
Prokhorov's theorem for countably many random measures on a Polish space
I am looking for help to show the following lemma:
Lemma Let $(\Omega,\mathcal A,\mathbb P)$ be a complete, standard Borel probability space and $\mathcal X$ a Polish space. Let $\mathcal P(\mathcal ...
6
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answers
220
views
Energy of harmonic maps from $\mathbb R^2$ to $S^2$ is quantized
Assume that $U:\mathbb R^2\to S^2=\{y\in\mathbb R^3:|y|=1\}$ is a smooth solution of the equation $\Delta U+|\nabla U|^2U=0$ in $\mathbb R^2$ with $\int_{\mathbb R^2}|\nabla U|^2\,dx<+\infty$. ...
- 517
6
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169
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Topological spaces for which $w(X)\leq |X|$ holds
Let $w(X) = \inf\{|\mathcal{B}| : \mathcal{B} \text{ is a base for }X\}$ be the weight of topological space $X$.
For metric spaces and locally compact spaces we have inequality $w(X)\leq |X|$.
This ...
- 1,201
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98
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Conditions for completely positive maps to act homomorphically across multiple subalgebras
For a completely positive (CP) map $u: A \to A'$ of $C^*$-algebras $A, A'$, the concept of multiplicative domains characterizes the largest subalgebra of $A$ on which $u$ behaves as a $*$-homomorphism....
- 61
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220
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Is the Taylor map continuous?
(Skip to the bolded theorem below for my question, if you'd like)
Some context on asymptotic expansions and the Taylor map
In the setting of irregular singularities of meromorphic connections on the ...
- 370
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votes
1
answer
331
views
If $t \to \lVert f(\cdot,t) \rVert_{L^2_x}^2$ is absolutely continuous, can we interchange the spatial integral and time derivative? (from MSE)
I originally posted this question on MSE. But it seems more nontrivial than expected, so I guess MO is a more appropriate place to ask.
I repeat the question for the sake of completeness:
Let $f(x,t) ...
- 3,477
6
votes
0
answers
201
views
Dependence of Neumann eigenvalues on the domain
I have the following problem in hands, in the context of a broader investigation:
Let $V\in L^{n/2}$ compactly supported, where $n\geq 3$ is the dimension. I want to prove the following:
For any $\...
- 101
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votes
0
answers
253
views
Are bounded groups of thin operators on Hilbert space similar to groups of unitaries?
QUESTION. Let $G$ be a group of bounded operators on $\ell^2$, satisfying $\sup_{x\in G} \lVert x\rVert <\infty$, whose elements are all of the form "identity+compact" (sometimes called &...
- 25.8k
6
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answers
182
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Conditions for metrisability
If a normal, first countable space is the union of countably many open metrisable subspaces, must that space be metrisable?
Partial answers, which I proved in the 1980's, include:
(0) The answer is ...
- 69
6
votes
0
answers
187
views
Gaussian lower heat kernel bounds on non-convex bounded domain
I am looking for a proof the following theorem.
Let $U \subset \mathbb{R}^n$ be a bounded domain with $C^2$ boundary and $p(x,y,t)$ be the Neumann heat kernel. Then there exist a constant $C>0$ ...
- 61
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0
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246
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Making the analogy of finiteness and compactness precise
If one asks about the intution behind compact topological spaces, most often one will hear the mantra
“Compactness of a topological space is a generalisation of the finiteness of a set.”
For example,...
- 1,474
6
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0
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155
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Metric spaces containing a topological disc
It is well-known that every connected, locally connected compact metrizable space $X$ contains an arc, that is, a subspace homeomorphic to $[0,1]$. Are there topological properties we can add to these ...
- 7,193
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217
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Detailed examples of induction on scale
I'm trying to understand the induction on scale argument in harmonic analysis. On this abstract it's mentioned that induction on scale can be used to prove Cauchy Schwartz inequality, Beckner's tight ...
- 265
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0
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111
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A generalized Hausdorff dimension in form of a Lower semi continuous function
Let $(X,d)$ be a compact metric space. Assume that $f:X\to \mathbb{R}$ is a positive continuous function. We say that the $f$-dimension of $(X,d)$ is equal to $0$ if for every $\epsilon>0$ ...
- 356
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123
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Can two eigenfunctions be almost linearly dependent in a region?
Consider the Schrödinger operator $H=-\Delta+|x|^a$ on $\mathbb{R}$, where $a>0$. Since the potential is growing at $\infty$, we have compact resolvent thus the eigenvalues are discrete and tend to ...
- 879
6
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188
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Measurability of eigenvalues-eigenvectors of a positive compact operator
Let $H$ be a separable Hilbert space over $\mathbb{R}$. Let ${A} = \{a\colon H\to H\,|\,a\text{ is a positive, compact linear operator}\}$.
By the spectral theorem, given $a \in A$, there are scalars $...
- 163
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255
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Every Polish space is the image of the Baire space by a continuous and closed map, reference
The following result was originally proven by Engelking in his 1969 paper On closed images of the space of irrationals (AMS, JSTOR, MR239571, Zbl 0177.25501)
Every Polish space (i.e. every separable ...
- 2,286
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0
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110
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Heat Flows and spatial singularities
While working on an abstract problem, I came up with the following question:
Let $\Omega_1 := \mathrm B(-1, 1)$ and $\Omega_2 := \mathrm B(1, 1)$, where $\mathrm B(x, r) \subseteq \mathbb R^2$ denotes ...
6
votes
0
answers
112
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Classification of contractible open n-manifolds which embed in a compact n-manifold
Does there exist a classification of contractible open $n$-manifolds ($n\geq 3$) which embed in a compact $n$-manifold? More general, does there exist a classification of contractible open $n$-...
- 2,083
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0
answers
309
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Have we discovered constructions for natural fractional dimensional spheres?
I have been thinking about a couple different problems in fractal geometry (including I one deleted because it was ill posed) and realize they all depend in a fundamental way on the problem of: Can we ...
- 2,689
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0
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131
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A theorem by R.L. Moore
The following result is due to R.L. Moore.
Let $K\subseteq\mathbb C$ be compact. Suppose that
$K$ is connected,
and that $\mathbb C\setminus K$ is connected.
Then $\partial K$ is connected.
Does ...
- 687