All Questions
13,927 questions
2
votes
1
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165
views
Hereditarily locally connected spaces
A topological space is locally connected if every point has a neighborhood basis of connected open subsets.
A property of topological spaces is termed hereditary, subspace-hereditary, if every subset ...
- 1,422
4
votes
0
answers
179
views
What's the unspoken history of compactly generated topological spaces?
Usually, the alleged motivation for the definition of compactly generated topological spaces is Cartesian closedness, which fails for general spaces. Of course, from a contemporary perspective, this ...
- 321
6
votes
3
answers
965
views
Convolution of $L^2$ functions
Let $u\in L^2(\mathbb R^n)$: then $u\ast u$ is a bounded continuous function. Let me assume now that $u\ast u$ is compactly supported. Is there anything relevant that could be said on the support of $...
- 24.2k
4
votes
1
answer
786
views
The definition of simple eigenvalue
This question was posted a long time ago on the mathexchange, but I didn't get any answers there, and despite having discussed it with some colleagues, I don't think I have a definitive answer.
I am ...
- 12.6k
4
votes
1
answer
161
views
For centered Gaussian measures, is $E[\lVert X\rVert^2] \lesssim E[\lVert X\rVert ]^2$ true in infinite dimensions as well?
In the proof of Corollary 5.7 in the following link:
https://arxiv.org/pdf/1610.05200.pdf
the author claims that $E[\lVert X\rVert^2] \lesssim E[\lVert X\rVert ]^2$ for the standard normal ...
- 128k
3
votes
2
answers
203
views
Recovering a set from its projections in varying coordinate systems - a projection hull?
Let me describe the simplest non-trivial case of what I have in mind. Let $V$ be a 2-dimensional $\mathbb{R}$-vector space and fix an isomorphism $V \cong \mathbb{R}^2$, where $\mathbb{R}^2$ is ...
- 55.9k
1
vote
0
answers
192
views
Simple left earthquakes are dense
i´ve been studying an article from W. P. Thurston about hyperbolic geometry, there, he defines something called left earthquake, whose definition is as follows:
Definition. If $\lambda$ is a geodesic ...
6
votes
1
answer
244
views
The continuity of certain maps on compact Hausdorff spaces
Let $f:M\to Y$ be a continuous proper bijective map from a metrizable space $M$ onto a $T_1$-space $Y$. The properness of $f$ means that for every compact subspace $K\subseteq Y$ the preimage $f^{-1}[...
- 42k
3
votes
1
answer
190
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Can the equation $1+z+z^q=z^n$ have multiple complex roots $z$?
It is proved here that the equation $1+z+z^2=z^n$ have no multiple complex roots.
Q. Let us consider the equation $1+z+z^q=z^n$ where $q$ and $n$ are natural numbers with $1<q<n$. Any ...
- 22.5k
2
votes
0
answers
204
views
Schrödinger representation of the Heisenberg group
Let $\Pi_{\lambda}$ be the the Schrödinger representations of the Heisenberg group $H^n=\Bbb C^n\times\Bbb R$. For $\phi\in L^2(\Bbb R^n)$, we have
$$\Pi_{\lambda} (x,y,t)\phi(\xi)=e^{i\lambda t} e^{...
- 531
2
votes
1
answer
112
views
The eigenvectors of adding a particular rank one matrix to the circulant matrix
Suppose that $e_1, \cdots, e_n$ are the standard vectors of the Euclidean space $\mathbb{R}^n$.
Let us consider the backward shift operator $T:\mathbb{R}^n\to \mathbb{R}^n$ given by $Te_k=e_{k-1}$ ...
- 128k
6
votes
2
answers
519
views
Existence of an integral equation (Faedo-Galerkin, Banach fixed point, Picard-Lindelof)
This question is concerning the paper, particularly the proof of Lemma 2.1 in Section 2.1:
Matas, A., Merker, J. Existence of weak solutions to doubly degenerate diffusion equations, Appl Math 57 (...
- 4,713
7
votes
7
answers
3k
views
Gelfand representation and functional calculus applications beyond Functional Analysis
I think it is fair to say that the fields of Operator Algebras, Operator Theory, and Banach Algebras rely on Gelfand representation and functional calculus in a crucial way.
I am curious about ...
- 4,713
1
vote
1
answer
131
views
A generalized form of the approximation to identity?
This question is an extension of the one I posted on ME: https://math.stackexchange.com/questions/4701500/if-alpha-nx-int-lvert-x-y-rvert-leq-1-n-lvert-x-y-rvert2-d-muy
It might be elementary for here,...
- 128k
0
votes
0
answers
307
views
Generalizations of the generalized Stokes theorem and the Atiyah-Singer index theorem
I am interested in the generalized Stokes theorem and its various generalizations. It is apparent to me that many theorems in vector analysis and certain theorems in complex analysis can be viewed as ...
- 63.9k
2
votes
0
answers
175
views
Banach space of vector measures
Let $S$ be a set and $\Sigma$ be a $\sigma$-algebra of subsets of $S$. Let $A$ be a Banach space over the field of complex numbers. A countably additive map $\mu:\Sigma\to A$ is called a vector ...
- 552
3
votes
0
answers
69
views
Is every weakly $1$-dimensional space embeddable in the plane?
A $1$-dimensional (separable metric) space $X$ is weakly $1$-dimensional if $$\Lambda(X)=\{x\in X:X\text{ is 1-dimensional at }x\}$$
is zero-dimensional (i.e. the space $\Lambda(X)$ has a basis of ...
- 3,317
3
votes
1
answer
148
views
positive functional on Banach *-algebra (with appro. identity) is continuous?
Theorem (N. Th.Varopoulos): Let $\mathcal{B}$ be a Banach *-algebra
with a bounded approximate identity. Then every positive functional $T$ on $\mathcal{B}$ is continuous.
I think this theorem is ...
- 2,545
5
votes
2
answers
1k
views
How to prove the second Korn inequality?
$\textbf{Theorem}.1$ (The first Korn inequality) Suppose that $ \Omega $ is a bounded domain in $ \mathbb{R}^d $ with Lipschitz boundary. Then\
\begin{eqnarray}
\sqrt{2}\left\|\triangledown u\right\|_{...
- 6,367
3
votes
1
answer
627
views
Compact embedding between parabolic Hölder spaces
My question is about the following compact embedding:
\begin{equation}
C^{\sigma+2, \sigma/2+1}_{x, t}(Q_T)\hookrightarrow C^{\sigma, \sigma/2}_{x, t}(Q_T).
\end{equation}
what condition should be put ...
CommunityBot
- 1
17
votes
1
answer
570
views
Does a completely metrizable space admit a compatible metric where all intersections of nested closed balls are non-empty?
(cross-posted from this math.SE question)
It is well-known that given a metric space $(X,d)$, the metric is complete if and only if every intersection of nested (i.e. decreasing with respect to ...
- 42k
1
vote
0
answers
103
views
"Classifying" causally closed sets in Minkowski space
Let $M = \mathbb R^{D+1}$ be Minkowski space. Recall that the causal complement of a set $A \subseteq M$ is the set $A^\perp \subseteq M$ where $p \in A^\perp$ there is no timelike path between $p$ ...
- 64k
1
vote
1
answer
160
views
On an integral equation
Let $B: C^{\infty}([0,1]^3)$ satisfy
$$B(t,t,x)=0 \quad \text{for all $t,x \in [0,1]$.}$$
Let $f \in C^{\infty}([0,1]^2)$ satisfy the following integral equation:
$$ \int_0^1 f(t,x)\,dx + \int_0^t\...
- 6,696
1
vote
0
answers
133
views
Does the Gaussian Poincare inequality hold for infinite dimensional measure metric spaces?
This is a question subsequent to the one:
Does the Gaussian Poincare inequality hold for $p=1$ as well as $p=2$?
There, I received a very helpful answer that the Gaussian poincare inequality for any ...
- 3,477
8
votes
0
answers
952
views
About generator and isomorphism problems for free groups operator algebras
Let $H$ be an infinite dimensional separable Hilbert space.
The $C^{*}$-algebras and von Neumann here unital and subalgebras of $B(H)$.
Definition : Let $\mathcal{A}$ be $C^{*}$-algebra (resp. a von ...
0
votes
0
answers
114
views
Norm distance in a Banach space
Consider the Hilbert space $l_2(\mathbb{N})$ under the square summable norm $\Vert \cdot \Vert_2.$ Let us define a new norm $||| \cdot ||| $ equivalent to $\Vert \cdot \Vert_2$ such that the closed ...
- 85
1
vote
0
answers
108
views
Existence of a smooth extension
In the three dimensional Euclidean space $\mathbb R^3$ let us define the hypersurface
$$ S= \{(x,y,z) \in \mathbb R^3\,:\, z^2= x^2+y^2\}.$$
Suppose that $f \in C^{\infty}(S)$. Does there exist $u\in ...
- 4,115
5
votes
1
answer
953
views
Comparison of two versions of fractional Sobolev spaces: do we have $W^{s,p}(\mathbb{R}^{n})=H^{s,p}(\mathbb{R}^{n})$?
There are two versions of fractional Sobolev spaces.
Definition 1: (Via Gagliardo semi-norm) Let $1\leq p\leq +\infty$, $0<s<1$ and let $\Omega\subseteq \mathbb{R}^n$ be an open set. The ...
- 1,101
47
votes
4
answers
4k
views
Which topological spaces admit a nonstandard metric?
My question is about the concept of nonstandard metric space that would arise from a use of the nonstandard reals R* in place of the usual R-valued metric.
That is, let us define that a topological ...
- 11.4k
1
vote
0
answers
79
views
Groups without "almost equivariant" coarse embeddings
Let $X$ be a set. We say that $\psi:X\times X\to[0,\infty)$ is a CND (conditionally negative definite) kernel if there is a Hilbert space $\mathcal{H}$ and a map $f:X\to\mathcal{H}$ such that
\begin{...
- 81
1
vote
1
answer
319
views
Total variation distance
Let $\mathcal{X}$ be the input or feature space, let $\mathcal{B}$ be Borel $\sigma$-algebra on $\mathcal{X}$ and $P(\mathcal{X})$ denotes the set of all probability measures on $(\mathcal{X},\mathcal{...
- 128k
1
vote
1
answer
177
views
B-topological ring that is not semi-topological?
A topological space $A$ that is also a ring with operations ‘$+$’ and ‘$.$’, is a semi-topological ring if the mappings
\begin{gather*}
a_1: A\times A \to A\text{ such that }(x,y)\to x+y, \\
a_2: A \...
- 12.9k
2
votes
0
answers
57
views
Regularity on $\mathbb{T}^3$ of the "functional average" of a map $S : C^\infty(\mathbb{T}^3, \mathbb{R}) \to L^2(\mathbb{T}^3, \mathbb{R})$
For simplicity, let $C^\infty(\mathbb{T}^3, \mathbb{R})$ be the real Frechet space of periodic smooth functions on $\mathbb{R}^3$. Here, $\mathbb{T}^3$ is the $3$-dimensional torus.
For a fixed ...
- 3,477
3
votes
0
answers
107
views
Does the pseudo-arc contain Erdős space?
The pseudo-arc is the unique hereditarily indecomposable chainable continuum.
The Lelek fan is the unique compact, connected subset of the Cantor fan (the cone over the Cantor set) with a dense ...
- 3,317
3
votes
1
answer
182
views
How to choose some $h$ so its Fourier transform supported in some set?
Suppose that $K=[-N, N]$ is some compact subset of $\mathbb R$, for some $N>2.$
Can we expect to choose $h$ such that $h=1$ on $K$ and the support of the Fourier transform of $\widehat{h}$ ...
- 79.9k
1
vote
2
answers
590
views
Inequality between coefficients of a polynomial and its supremum
For $d, m \in\mathbb{N}$ fixed, let $P\equiv P(x) := \sum_{|\alpha|\leq m} c_\alpha\cdot x^\alpha$ be a real polynomial in $d$ variables of (total) degree $m$. (That is, the above sum ranges over all ...
- 11
2
votes
1
answer
390
views
Can we interpret fractional Sobolev spaces in terms of fractional derivatives?
Let $1 \leq p < \infty$, $0<s<1$, and $\Omega \subseteq R^n$ be a domain. The Banach space $W^{s,p}(\Omega)$ is defined as
$$W^{s,p}(\Omega) := \left\{ f \in L^p(\Omega) \colon \int_{\Omega \...
- 981
7
votes
3
answers
1k
views
Connected space being not locally connected at each point
Say that a topological space $X$ is locally connected at some point $x$, if it has a local base at that point consisting of connected open sets.
Also $X$ is locally connected if it is locally ...
- 351
8
votes
3
answers
1k
views
Fourier dimension of the sum of sets
This question came up when my supervisors, my colleague, and I were considering arithmetic progressions in sets of fractional dimension. In particular, we were interested in "extracting" Salem sets ...
0
votes
1
answer
154
views
Finite dimensionality of a subspace
Let $c>0$ and let $Y$ be the space of all distributions of compact support in $(-1,1)$ with singular support at $\{0\}$. Let $X$ be subspace of $Y$ such that for any $\phi \in X$ there holds:
$$ \...
- 128k
2
votes
0
answers
62
views
On a real smooth version of white noise distribution theory
In white noise analysis, one starts with a real Gelfand triple $\mathcal{N}\subset \mathcal{H} \subset \mathcal{N}^{*}$ and produces out of it, using complexifications along the way, the complex ...
- 577
7
votes
1
answer
262
views
Can you remove a zero dimensional subspace from a cube and obtain a planar space?
The question, which came up in a conversation with my advisor Ola Kwiatkowska, is pretty much in the title:
Let $Z\subseteq[0,1]^3$ be zero-dimensional. Is it possible for $[0,1]^3\setminus Z$ to be ...
- 800
1
vote
0
answers
105
views
Friedrichs extension of the Laplacian from a smooth subspace and density of its eigenbasis in the Frechet topology of the subspace as well?
Let $C^\infty_\text{div}(\mathbb{T}^3)$ be the "real" Frechet space of periodic, divergence-free smooth vector fields on $\mathbb{R}^3$. That is, $\mathbb{T}^3$ is the $3$-dimensional torus.
...
4
votes
0
answers
114
views
Find at least one square-boxed subcontinuum
Recall that a plane continuum is a closed, bounded,
connected subset of the plane.
It is non-degenerate if it contains at least two points.
(We may sometimes just say "continuum" even if
we ...
- 1,375
3
votes
1
answer
185
views
Is the weighted shift strong frequently hypercyclic?
One sided Shift
Let be $M$ separable metric space. Consider $X=M^{\mathbb{N}}$ the sequence space equipped with the product metric $d(x,y)=\sum_{i=1}^\infty |x_i-y_i|/2^i$ . Define the shift map $\...
- 146
1
vote
3
answers
619
views
Given a compact set $K \subset \mathbb{R}^n$, is the space of distributions supported on $K$ the dual of some test function space?
I am aware that the dual of $C^\infty(\mathbb{R}^n)$ is the space of distributions (not necessarily tempered) with compact support.
However, if we fix a compact set $K \subset \mathbb{R}^n$, is the ...
- 16.5k
4
votes
0
answers
197
views
Bounded cohomology and unitary representations
On page 9 of Nicolas Monod's very nice ICM report "An invitation to bounded cohomology" (https://egg.epfl.ch/~nmonod/articles/icm.pdf), he mentions that bounded cohomology may be related to ...
- 2,545
0
votes
0
answers
119
views
Weak convergence in $W^{ 1,q}(\Omega) $ $ (1 \leq q < \infty)$ space
Let $\Omega \subset \mathbb { R } ^ n $be a bounded domain, and suppose that in the space $W^{ 1,q}(\Omega) $ $ (1 \leq q < \infty)$, the sequence $\{ u_j \} $ converges weakly in $W^{ 1,q}(\Omega)...
- 18.7k
22
votes
1
answer
712
views
Are $\beta \mathbb{Q}$ and $\beta(\beta\mathbb{Q}\setminus\mathbb{Q})$ homeomorphic?
The canonical inclusion $\beta\mathbb{Q}\setminus \mathbb{Q} \hookrightarrow \beta\mathbb{Q}$ is not the Stone-Čech compactification of $\beta\mathbb{Q}\setminus \mathbb{Q}$. Even so, this doesn't ...
- 12.9k
4
votes
0
answers
495
views
Fixing the duality $L^\infty(X)= L^1(X)^*$ for Radon measure spaces
Consider the following fragment from Folland's book "A course in abstract harmonic analysis":
Let me denote the Borel subsets of $X$ by $\mathscr{B}(X)$. Folland claims that if $\mu$ is a ...
- 175