All Questions
13,927 questions
0
votes
1
answer
109
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Extending maps from a discrete set to a Stone-Čech compactification while retaining an injectivity condition
For $S$ a set, let $\beta_{\bf2}(S)$ be a compact, totally disconnected space containing $S$ where $S$ in the subspace topology is discrete and $S$ is a dense subspace, and $\beta_{\bf2}(S)$ has the ...
- 11.4k
3
votes
2
answers
285
views
Cut a homotopy in two via a "frontier"
Consider a space $G$ obtained by glueing two disjoint cobordisms (the fact that they are might be irrelevant, assume they are topological spaces at first) $L$ and $R$ on a common boundary $C$.
(...
- 52.7k
6
votes
1
answer
652
views
Nuclear spaces and intuition behind their topology
In functional analysis the nuclear spaces (coined by Grothendieck before he became involved in revolutionizing algebraic geometry) can be considered
as a kind of generalization of finite dimensional ...
- 111
1
vote
1
answer
69
views
Local completion of bornological space
I recently stumbled across this old publication which defines the local completion of a Hausdorff locally convex space.
The construction is as follows:
A Hausdorff locally convex space $E$ is locally ...
- 661
6
votes
1
answer
754
views
Banach Manifold
Let $M$ and $N$ be closed manifolds. Is it true that
$C^{k}(N,M)$, which is the space of functions $f: N\to M$ such that $f\in C^{k}$, is a $C^{\infty}$ Banach manifold? If so, can you help me to ...
- 2,821
20
votes
3
answers
3k
views
Realizing universal $C^*$-algebras as concrete $C^*$-algebras
How do I in general realize a universal C*-algebra generated by some generators and relation as concrete C*-algebras? For example, I know that universal C*-algebra generated by a single unitary is $C(\...
- 2,821
0
votes
0
answers
119
views
About definition of stable solution. $Q_u(\phi) \ge 0$ for all $\phi \in C_c^1(\Omega)$ replaced by "for all $\phi \in W_0^{1,2}(\Omega)$"
I want to ask about a remark about the stable solution of elliptic PDE Remark 1.1.1.
We say $u$ is stable solution of $-\Delta u=f(u) \ \text { in } \Omega$ and $u=0$ on $\partial \Omega$ if it ...
- 809
23
votes
2
answers
7k
views
What is a Gaussian measure?
Let $X$ be a topological affine space. A Gaussian measure on $X$ is characterized by the property that its finite-dimensional projections are multivariate Gaussian distributions.
Is there a direct ...
- 34.7k
5
votes
1
answer
615
views
Is every character of the algebra of continuous functions on a locally compact space some evaluation?
Given any locally compact Hausdorff space $X$, let $C(X)$ denote the complex algebra of all complex-valued continuous functions on $X$.
Question. Given an arbitrary character (i.e. a non-zero ...
- 12.9k
1
vote
0
answers
119
views
Continuity of a minimizing measure w.r.t a parameter
Let $V_t(x)=x^2+t\phi(x)$ where $t>0$ and $\phi\in C^\infty_c(\mathbb{R})$.
My question is what can be said about the continuity of the (unique) minimizer (among probability measures) of the ...
- 41
11
votes
2
answers
721
views
Spherical harmonics – pointwise and L1 bounds
Let $\{ \phi _{d,m}\}_{m\geq 1}$ be multi-dimensional spherical harmonics, i.e., solutions of $\Delta \phi = E\phi$ on the sphere $S^d$ for $d>1$, arranged in an increasing order $E_1 \leq E_2 \leq ...
4
votes
1
answer
792
views
How to show that this limit converges in the distributional sense to a dirac delta function
Let
$$\begin{eqnarray}\nonumber
f(y, t) &=& \frac{C}{\sigma ^2 t} \left[\frac{(1-\alpha) (b-y)}{\alpha t^{\alpha}} \, _1F_1\left[\frac{\alpha+1}{2 \alpha};\frac{3}{2};-\frac{ (b-y)^2}{2 \...
4
votes
1
answer
291
views
Structure of all Wightman QFTs
I have two related questions related to constructive/axiomatic QFT.
Is there a structure on the collection of all QFTs, as defined by the Wightman axioms? Do they form some type of category?
...
- 21.5k
5
votes
0
answers
131
views
Is the opposite of the category of $\kappa$-Lindelöf Hausdorff spaces locally presentable?
Gelfand duality tells us that the category of compact Hausdorff spaces (with continuous maps as morphisms) is contravariantly equivalent to the category of commutative, unital $C^\ast$-algebras (with $...
- 64k
0
votes
0
answers
155
views
Reducing subspaces of unitary operators
Let $U$ be a unitary operator on a Hilbert space $\mathcal{H}$. We can assume $\mathcal{H}$ is an $L^2$ space and $U$ acts as multiplication by a function $u$ with $|u(x)| = 1$ a.e (by the spectral ...
- 63.9k
1
vote
1
answer
120
views
Sobolev-type estimate for irrational winding on a torus
Let $\mathbb{T} = \{ (x, y) \in \mathbb{R}^2 \}/_{x \mapsto x + 1, y \mapsto y + 1}$ be a real 2-torus. Let $\mathscr{C}^{\infty}_0(\mathbb{T})$ be the subset of $\mathscr{C}^{\infty}(\mathbb{T})$ of ...
- 227
5
votes
1
answer
543
views
If $f$ is bounded, decays fast enough at infinity and $\int f=0$, does this imply that $f$ is in the Hardy space $\mathcal H^1(\mathbb R^n)$?
Let $\mathcal H^1(\mathbb R^n)$ be the real Hardy space (as in Stein's "Harmonic Analysis", Chapter 3). It is well known that $\mathcal H^1(\mathbb R^n)\subset L^1(\mathbb R^n)$ and its ...
- 1,075
1
vote
1
answer
344
views
Is there anyway to formulate the Alexandrov topology algebraically?
One knows that the Alexandrov topology on a preordered set is the finest topology that induces the same [specialization] preorder on the set.
Given this, one finds a one-to-one correspondence between ...
- 612
10
votes
0
answers
242
views
Arhangel'skii's problem revisited
One of the most well-known problems in set-theoretic topology is Arhangel'skii's question of whether there exists a Lindelöf Hausdorff space with "points $G_\delta$" (meaning, every point is ...
- 4,413
2
votes
0
answers
188
views
Self-adjointness of fractional laplacian
Lets consider the following functional analytic definition of the fractional Laplacian: Consider a (complete, connected, oriented) Riemannian manifold $(M,g)$ with corresponding Laplacian $\Delta_{g}$....
- 1,171
1
vote
1
answer
159
views
Do completely bounded maps on an operator space have a completely contractive Banach algebra structure?
Let $X$ be an operator space and $CB(X)$ be the set of all completely bounded linear maps $f: X \to X$. Note that $CB(X)$ becomes a Banach algebra for the composition of operators.
Is the ...
- 25.8k
8
votes
1
answer
380
views
Given an embedded disk in $\mathbb{R}^n$, is there always another disk which intersects it nontrivially in a disk?
We call an open subset $D\subset X$ of a manifold $X$ an embedded disk, if there exists a homeomorphism $D\cong \mathbb{R}^n$.
The precise formulation of the question in the title is as follows:
Let $...
- 12.3k
1
vote
0
answers
84
views
Is there a standard name for the following class of functions on non-Hausdorff manifolds?
Let $M$ be a (not necessarily Hausdorff) smooth manifold. Given an open chart $U\subset M$ and a compactly-supported smooth function $f:U\to\mathbb{R}$ on $U$, define $\widetilde{f}:M\to\mathbb{R}$ by ...
- 5,429
2
votes
0
answers
320
views
Why is a certain projective limit of weighted symmetric Fock space, namely $\bigcap\limits_{\tau \in T, p\ge 1 } \mathcal{F}(H_\tau,p)$, separable
I have a question regarding separability of a certain locally convex space.
Let $H_{\tau}:=H^{\tau_1}(\mathbb{R}^n,\tau_2(x)dx)$ the weighted Sobolev Hilbert space with $\tau_1 \in \mathbb{N}, \tau_2(...
6
votes
1
answer
310
views
Surjectivity of a class of integrals in dimensions two
Let $\Omega \subset \mathbb{R}^2$ be an open set and $G(x,\theta): \Omega \times [0,2\pi]\rightarrow \mathbb{R}$ be a positive continuous function. Assume $F:\Omega \rightarrow \mathbb{R}^2$ defined ...
- 1,075
92
votes
3
answers
14k
views
Is every sigma-algebra the Borel algebra of a topology?
This question arises from the excellent question posed on math.SE
by Salvo Tringali, namely, Correspondence
between Borel algebras and topology.
Since the question was not answered there after some ...
2
votes
1
answer
165
views
Continuity of an upper semi-continuous function over periodic points
Let $f: X \to \mathbb{R}$ be an upper semi-continuous function on $X$, which is a compact subspace of a vector space. Let sequence $x_n, n \in \mathbb{N}$, with positive elements - periodic: there ...
- 6,321
1
vote
1
answer
732
views
Notations for open and closed sets
I am wondering why a standard notation for open sets is $G$ and that for closed sets is $F$. I mean, $F$ precedes $G$ in the alphabet, whereas open sets are usually introduced before closed ones.
- 4,786
1
vote
1
answer
115
views
What is the socle of the $2\times 2$ matrix algebra over a Banach algebra?
$\DeclareMathOperator\soc{soc}$Let $\mathcal{A}$ be a unital semisimple Banach algebra. The socle of $\mathcal{A}$, $\soc(\mathcal{A})$ is defined as the sum of the minimal right ideals (which equals ...
- 63.9k
2
votes
1
answer
225
views
Boundary points in $\overline{\operatorname{conv}\{z_i\}_{i\in I}}$
Let $X$ be an infinitely-dimensional Banach space and $\{z_i\}_{i\in I}$ be a set of linearly independent points in $X_{\leq 1}$, the closed unit ball of $X$. $I$ the index set is not necessarily ...
- 16.5k
9
votes
2
answers
3k
views
Compact Hausdorff spaces without isolated points in ZF
$S$ is uncountable := $\vert\mathbb{N}\vert<\vert S\vert$
$S$ is noncountable := $\vert S\vert \not\leq \vert\mathbb{N}\vert$
$(X,T)$ is a nice space := $(X,T)$ is a compact Hausdorff space ...
- 2,031
19
votes
3
answers
1k
views
What standard Banach space is isomorphic to the completion of this different normed structure on $\ell^1$?
A colleague asked me the following question:
"What can one do with the following norm on $\ell^1$: $|x|=\int_1^2 |x|_pdp$ where $| \;\; |_p$ is the standard norm on $\ell_p$?"
This ...
- 1,075
2
votes
1
answer
150
views
Sufficient conditions for the graph measurability of a multivalued function
I am currently working on a problem related to the measurability of multi-functions in the context of mathematical economics. Specifically, I am searching for sufficient conditions regarding the graph ...
2
votes
0
answers
329
views
Conditions for an existence of smooth solution to a parabolic PDE
I'm interested to know the conditions of when the parabolic PDE ($U \subset \mathbb{R}^n$ is some bounded open subset):
\begin{equation*}
u_t - \sum_{i,j=1}^n(a^{ij}(x,t)u_{x_i})_{x_j} + \sum_{i=1}^nb^...
- 6,367
2
votes
1
answer
141
views
Is there a bound on the number of connected components of a zero set of an integrable function?
If $f$ is a real-analytic function on $[0,1]^n$, and $f$ has finite differential transcendence degree, is there some way to bound the number of connected components of its zero set or the set where it ...
- 1,446
2
votes
0
answers
157
views
Why do von Neumann algebras possess identity?
My starting point is that a von Neumann algebra is a $C^*$-algebra with a predual. The usual approaches to showing the existence of identity involve spectral theory (for approximate identity), but ...
- 433
0
votes
0
answers
73
views
Can we construct general counterexample to support the Weak Whitney theorem? [duplicate]
Can we construct an example for the weak Whitney theorem to illustrate the existence of a continuous function from an $n$-dimensional manifold to an $m$-dimensional manifold that cannot be smoothly ...
- 6,367
1
vote
0
answers
131
views
Can we construct a general counterexample to support the weak whitney embedding theorm?
The weak Whitney embedding theorem states that any continuous function from an $n$-dimensional manifold to an $m$-dimensional manifold may be approximated by a smooth embedding provided $m > 2n$.
...
- 109
3
votes
1
answer
198
views
Do radially bounded sets form a bornology?
We call a subset $A$ in a real vector space $E$ radially bounded if it intersects every ray emanating from $0$ via a bounded set. It is easy to see that radially bounded sets in $E$ form a bornology, ...
- 6,367
3
votes
1
answer
298
views
Pointwise convergence and disjoint sequences in $C(K)$
Let $K$ be a Hausdorff compact space and let $C(K)$ be the space of continuous real-valued functions on $K$. A sequence $(h_n)$ in $C(K)$ is called almost disjoint if there is a sequence $(g_n)$ with ...
CommunityBot
- 1
3
votes
0
answers
200
views
Contractibility of the pseudo-boundary of the Hilbert cube
Let the separable Hilbert cube $Q=\prod_{i=1}^{+\infty}[0,1]$ embed into the real Hilbert space $H=l^2(\mathbb{Z}^+)$, whose coordinate unit vectors are $\{ e_i \}_{i=1}^{+\infty}$, as the subset $\...
- 1,543
6
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
4
votes
2
answers
378
views
A possible measure-theoretic pathology
Let $S$ be a nonempty closed subset of the open unit square $(0,1)^2 = X \times Y$
that has the following "shadow property":
For any aligned open square $C = A \times B$ that intersects $S$, ...
- 17.2k
0
votes
0
answers
162
views
Gluing faces of n-cube
Assuming $C_n$ be the $n$-cube, the intersection of $C_n$ with a supporting hyperplane $H(P, v)$ is called a face or more precisely a $d$-face if the dimension is $d$.
Let $f_0$ and $f_1$ be faces ...
- 4,589
1
vote
0
answers
54
views
Minimal F-semi-norms
There are conflicting terminologies in the literature on this subject, so let me define an F-semi-norms on a real vector space $E$ to be a subadditive function $\rho:E\to[0,+\infty)$ such that $\rho\...
CommunityBot
- 1
0
votes
2
answers
160
views
Bounded inverse to morphism of Banach algebras
Let $A:X\to Y$ be a surjective morphism of Banach spaces.
1) Does there always exists $B_R$, a bounded right inverse to $A$?
2) Assume additionally that $A$ is a morphism of unital Banach algebras. ...
- 2,821
2
votes
0
answers
180
views
Approximating $L^p$ functions by eigenfunctions of Laplacian
I'm reading a paper https://www.sciencedirect.com/science/article/pii/S0022039608004932.
In this paper, the authors assume that $\mathcal{O}$ is a bounded domain of $\mathbb{R}^N$ with $C^m$ boundary ...
- 121
7
votes
1
answer
654
views
Extending Hölder functions
I originally asked this question on MathStackExchange some time ago, but it seems that MathOverflow would be more appropriate. Essentially, I would like to find references for extension theorems for (...
- 6,367
1
vote
1
answer
143
views
$L^1$ error between indicator function and smoothed out version
For a large parameter $r>0$, consider the indicator function $1_{[-r,r]}$ and its convolution with the (normalized) Gaussian $\frac{1}{\sqrt{\pi}}e^{-x^2}$, that is,
$$f_r(x) = \frac{1}{\sqrt{\pi}}\...
- 24.2k
1
vote
1
answer
84
views
optimization over moving domains
Let $A, B$ be Banach spaces, and for any $a\in A$, $B_a\in B$ is a measurable subset. Consider the following optimization problem:
$$L(a)=\inf_{b\in B_a}\ell(b),$$
where $\ell(b)$ is a infinite-times ...
- 128k