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12 votes
1 answer
393 views

Is $X\times X$ homeomorphic to $X$ for a space of probability measures?

Let $\mathcal M_1(S)$ be the (compact, metrizable) space of probability Borel measures on the circle $S=\{z\in\mathbb C: |z|=1\}$ with its weak $*$ topology, so $\mu_n\to\mu$ if and only if $$ \int_S ...
Christian Remling's user avatar
1 vote
0 answers
104 views

Commutative Banach $\mathbb{R}$-algebras without complex structure, but with path-connected group of units

For a finite-dimensional commutative (associative, unital) $\mathbb{R}$-algebra $A$, the condition $\pi_0(A^\times) = 1$ (i.e. the group of units of $A$ being path-connected) is equivalent to $A$ ...
M.G.'s user avatar
  • 7,127
2 votes
2 answers
154 views

Closure of $C([0,1]^2)$ via weak*-topology [closed]

Let $C([0,1]^2)$ denote the set of continuous functions on $[0,1]^2$. Let $L^1([0,1]^2)$ be the set of all Lebesgue integrable functions on $[0,1]^2$. The dual space of $C([0,1]^2)$, denoted by $C^*([...
tom jerry's user avatar
  • 349
0 votes
1 answer
231 views

Questions on the compactness of $L_1([0,1]^2)$'s unit sphere

Let $U$ denote the set of functions $f\in L_1([0,1]^2)$ such that $\int f=1$ and $f(x,y)\geq 0: a.e. (x,y)\in [0,1]^2$. Recently in my study I need to study the compactness of $U$. By Riesz's theorem ...
tom jerry's user avatar
  • 349
0 votes
1 answer
101 views

Limit sequence of regular function in $L_1$‘s unit sphere

Let $U$ denote the set of functions $f\in L_1([0,1]^2)$ such that $\int f=1$. For any $f\in U$, we say it is regular if $\int_{x_0\times [0,1]}f=\int_{[0,1]\times y_0}f=1$ for a.e. every $x_0, y_0\in [...
tom jerry's user avatar
  • 349
-1 votes
1 answer
167 views

Space of distributions on $[0,1]^2$: weakly compact or not?

Let $X_1,X_2$ be distributions on $[0,1]$ and let $X=(X_1,X_2)$ be the joint distribution of $X_1,X_2$. Let $\mathcal{X}$ be the set of all such joint distribution $X$. Question 1: Does $\mathcal{X}$ ...
tom jerry's user avatar
  • 349
2 votes
1 answer
49 views

Is any submetrizable linear topology linearly submetrizable?

Let $E$ be a vector space. A topology $\tau$ on $E$ is called (linearly) submetrizable if there is a (linear) metrizable topology $\pi$ on $E$ which is weaker than $\tau$, i.e. $\pi\subset\tau$. Is ...
erz's user avatar
  • 5,529
1 vote
0 answers
87 views

Convergence and sequential compactness for nonlinear operators

I have a family of operators $T_n\colon X \to Y$ where $X,Y$ are Hilbert spaces. These operators are nonlinear. What kind of notions of convergence does one have for such operators? I'm specifically ...
C_Al's user avatar
  • 251
1 vote
2 answers
220 views

A question on unit norm elements of $\ell^2 \setminus \bigcup_{0<p<2 }\ell^p$

Let $A\subset \ell^2$ consist of all $x\in \ell^2$ with $|x|_2=1$ which does not belong to any $\ell^p$ for all $0<p<2$. Note that $A$ is non-empty with a Baire category argument. I ...
Ali Taghavi's user avatar
3 votes
1 answer
177 views

Compactness of set of measurable functions between compact subspaces of real numbers

Let $X$ be a compact subset of $\mathbb{R}^n$ and $Y$ be a convex and compact subset of $\mathbb{R}^p$. Consider $\mathcal{F}$ the set of all measurable functions from $X$ to $Y$. Can I find ...
guest1's user avatar
  • 131
15 votes
1 answer
601 views

Topological spaces in which countable intersections of dense open sets have dense interior

In certain topological spaces, known as Baire spaces (e.g., completely metrizable spaces), a countable intersection of dense open sets is dense. Now consider the following strengthening of the Baire ...
Gro-Tsen's user avatar
  • 32.5k
0 votes
0 answers
72 views

Sequential compactness via Arzela-Ascoli theorem for uniform state spaces

Let $X$ be a uniform topological space and $C([0,1],X)$ the space of continuous functions from [0,1] to $X$. Assume that for subsets of $X$ sequential compactness and compactness are equivalent. Let $(...
PDEprobabilist's user avatar
2 votes
0 answers
81 views

Extension of a tangent vector field

Let $\Omega$ be an open subset of $S^2$ with $\overline{\Omega} \neq S^2$. Suppose a continuous tangent vector field $G$ is defined on $\partial \Omega$ such that $|G(y)| = 1$ for all $y \in \partial \...
MathLearner's user avatar
4 votes
0 answers
108 views

Larger possible chain of closed subspaces in the dual of a Banach space

In this question, is demonstrated that a separable space can have a chain (ordered by inclusion) of closed subspaces with uncountable many subspaces. My question is the following. If $X$ is an ...
Emerick's user avatar
  • 153
2 votes
1 answer
79 views

Hausdorff-Lipschitz continuity of cone correspondence

Let $\mathbb{R}_+$ denote the strictly positive real numbers, let $\mathcal{X} \subset \mathbb{R}^n$ and $\mathcal{P} \subset \mathbb{R}^m$ be compact and convex subsets, let \begin{equation} f: \...
Heinrich A's user avatar
0 votes
1 answer
127 views

Continuous extensions of tangent vector fields

Let $\Omega$ be an open subset of $S^2$ with $\bar{\Omega}\neq S^2$. Suppose a continuous tangent vector field $G$ is given on $\partial \Omega$ with $|G(y)|=1$ for all $y\in \partial \Omega$. Does ...
MathLearner's user avatar
0 votes
1 answer
80 views

Continuous modification of tangent vector fields

Let $\Omega$ be an open subset of $S^2$, and assume that there exists a continuous tangent vector field $F(x)$ defined on $\bar{\Omega}\neq S^2$ with $|F(x)|=1$ for all $x\in \bar{\Omega}$. Suppose a ...
MathLearner's user avatar
0 votes
0 answers
97 views

Generator of an analytic semigroup

Perhaps I have a naive question. My question is as follows: When we consider a Cauchy proposition of the following form: $$ \begin{cases} x'(t)= -Ax(t)+ F(t,x(t)) &\text{for}\ t> 0 \\ x(0)=...
Mathlover's user avatar
1 vote
0 answers
76 views

Uniform approximation over compacts using weighted function spaces

I'm interested in approximations over the so-called weighted function spaces. Let $(X,\tau_X)$ be some completely regular Hausdorff topological space. Additionally, consider some map $\psi: X \to (0,\...
Gaspar's user avatar
  • 161
9 votes
2 answers
471 views

Proving the inequality involving Hausdorff distance and Wasserstein infinity distance

Prove the inequality $$d_{H}(\mathrm{spt}(\mu),\mathrm{spt}(\nu))\leq W_{\infty}(\mu,\nu)$$ where $d_H$ denotes the Hausdorff distance between the supports of the measures $\mu$ and $\nu$, and $W_\...
Luna Belle's user avatar
9 votes
1 answer
428 views

The cardinality of projections of subsets of the Hilbert cube by inner products

I have three related questions. Question 1: Is there a subset $X$ of the Hilbert cube $[0,1]^{\Bbb N}$ of cardinality continuum, such that for each sequence $a\in [0,1]^{\Bbb N}$ with $\sum a_n$ ...
Boaz Tsaban's user avatar
  • 3,104
2 votes
0 answers
35 views

Continuity of Kernel Mean Embeddings

Given some kernel $k: X \times X \to \mathbb{R}$ with RKHS $H_k$ we say that $k$ is characteristic on the space of signed Radon measures over $X$, denoted by $\mathcal{M}(X)$, if the kernel mean ...
Gaspar's user avatar
  • 161
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, \, \...
Julian Hölz's user avatar
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\...
Pietro Majer's user avatar
  • 60.5k
11 votes
1 answer
308 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 ...
Pietro Majer's user avatar
  • 60.5k
1 vote
1 answer
152 views

Points in the Stone Cech compactification are intersection of open sets

Let $\beta \mathbb{N}$ be the Stone Cech compactification of the natural numbers and let $ x\in \beta \mathbb{N}$. Is it true that there exists a sequence of open sets $\{U_n\}_{n=1}^\infty$ in $\beta ...
Serge the Toaster's user avatar
0 votes
1 answer
192 views

A continuous injection from the Hilbert cube to the real line?

Continuing an earlier "too good to be true" question that I posted recently, the same holds for the present question: Is there a continuous injection from the Hilbert cube $[0,1]^{\Bbb N}$ ...
Boaz Tsaban's user avatar
  • 3,104
1 vote
1 answer
162 views

Is there a uniformly continuous injective image of $(0,1)\setminus\Bbb Q$ in the Cantor space?

It seems too good to be possible, but: Is there a uniformly continuous injective image of $(0,1)\setminus\Bbb Q$ in the Cantor space? Here, the Cantor space $\{0,1\}^{\Bbb N}$ is equipped with the ...
Boaz Tsaban's user avatar
  • 3,104
0 votes
0 answers
98 views

Does weak $L^2$ approximation implies $L^2$ approximation under a condition similar to convexity?

(Cross posted from Math StackExchange: Does weak $L^2$ approximation implies $L^2$ approximation under a condition similar to convexity?) Assume $(\Omega, \mu)$ is a probability space. Consider a ...
David Gao's user avatar
  • 2,830
4 votes
1 answer
177 views

Compact-open Topology for Partial Maps?

I asked the same question on MathStackExchange a month ago and received no answer. I feel that this would be more suitable for MathOverflow. Compact open topology is one of the most common ways of ...
Bumblebee's user avatar
  • 1,093
3 votes
0 answers
239 views

Metrizing pointwise convergence of *sequences* of functionals in a dual space

This question was asked by myself on the math stackexchange a few days ago. I thought I'd repeat it here: Let $X$ be a normed, real vector space of uncountable dimension. Let $X^*$ denote the set of ...
Mustafa Motiwala's user avatar
0 votes
1 answer
92 views

Continuous selectors of a continuous multifunctin on a compact metric space

I am currently working on a continuous selector problem of multifunctions. I am trying to figure out if a continuous multifunction defined on a compact metric space always admit a continuous selector. ...
Saito's user avatar
  • 79
2 votes
1 answer
334 views

Hahn-Banach theorem and ultrafilter lemma

I'm unable to understand a remark in "Two application of the method of construction by ultrapowers to analysis" by Luxemburg, which uses the ultrafilter lemma to prove the Hahn-Banach ...
oggius's user avatar
  • 95
3 votes
1 answer
161 views

Approximating continuous functions from $K\times L$ into $[0,1]$

Let $K$ and $L$ be compact Hausdorff spaces, let $f:K\times L\to [0,1]$ be continuous and let $\varepsilon>0$. Can we find continuous $g_{1},...,g_{n}:K\to[0,+\infty)$ and $h_{1},...,h_{n}:L\to[0,+\...
erz's user avatar
  • 5,529
11 votes
1 answer
341 views

Density of linear subspaces in $C(K)$

Let $K$ be a compact Hausdorff space and denote by $C(K)$ the space of all real valued and continuous functions on $K$. We endow $C(K)$ with the supremum norm topology, making it a Banach space. ...
Julian Hölz's user avatar
7 votes
0 answers
150 views

The space of analytic associative operations

This question is a follow-up to this old one of mine. Let $\mathcal{A}$ be the set of functions $\star:\mathbb{R}^2\rightarrow\mathbb{R}$ which are associative and $C^\omega$ (real analytic entire) in ...
Noah Schweber's user avatar
8 votes
0 answers
192 views

Is $L^2(I,\mathbb Z)$ homeomorphic to the Hilbert space?

I am somehow puzzled by the subset $G:=L^2(I,\mathbb Z)$ of $H:=L^2(I,\mathbb R)$ of all integer valued functions on $I=[0,1]$ (in fact I mentioned as an example in this old MO question). Some simple ...
Pietro Majer's user avatar
  • 60.5k
2 votes
1 answer
264 views

Is a continuous functional on continuous functions the restriction of a continuous functional on the space of all functions?

As sets, we can consider the space $C(\mathbf{R}^n;\mathbf{R}^k)$ - of all continuous functions from $\mathbf{R}^n$ to $\mathbf{R}^k$ - to be a subset of the product space $(\mathbf{R}^k)^{\mathbf{R}^...
SBK's user avatar
  • 1,179
2 votes
1 answer
223 views

Is the projective limit $\mathcal{D}(\mathbb{R})$ separable?

Let $\mathscr{D}(\mathbb{R})$ be the set $C_0^\infty(\mathbb{R})$ of smooth functions with compact support endowed with the following topology: The initial topology with respect to the family maps $(\...
CoffeeArabica's user avatar
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 ...
erz's user avatar
  • 5,529
2 votes
1 answer
155 views

Variation of concept of a Lusin space

Citing from Wikipedia, A Hausdorff topological space is a Lusin space if some stronger topology makes it into a Polish space. Is there a (previously studied) analogous concept of a Hausdorff (...
iolo's user avatar
  • 651
1 vote
0 answers
73 views

Is $L^p_\text{loc} (Y)$ dense in $(L^0(Y), \hat \rho)$?

Below we use Bochner measurability and Bochner integral. Let $(Y, d)$ be a separable metric space, $\mathcal B$ Borel $\sigma$-algebra of $Y$, $\nu$ a $\sigma$-finite Borel measure on $Y$, $(Y, \...
Analyst's user avatar
  • 657
1 vote
1 answer
119 views

Extremally disconnected rigid infinite Hausdorff compacta(?)

Question: does there exist an extremally disconnected infinite Hausdorff compact space $\ X\ $ such that the only homeomorphism $\ h: X\to X\ $ is the identity homeomorphism $\ h=\mathbb I_X:\ X\to X\...
Wlod AA's user avatar
  • 4,786
1 vote
1 answer
74 views

Subspaces generated by the orbits of the group of isometries on $C(K)$

Let $X$ be an extremally disconnected compact Hausdorff space with no open points, and $f:X\to\mathbb{C}$ be a non-constant continuous function. Let $D_f$ be the linear span of the functions of the ...
Onur Oktay's user avatar
  • 2,605
0 votes
1 answer
253 views

Is the space $L^p_{\text{loc}} (\mathbb R^d)$ separable w.r.t. the norm $\|f\|_{\tilde L^p} := \sup_{x \in \mathbb R^d} \|1_{B(x, 1)} f\|_{L^p}$?

Fix $p \in [1, \infty)$. Let $(L^p (\mathbb R^d), \|\cdot\|_{L^p})$ be the Lesbesgue space of $p$-integrable real-valued functions on $\mathbb R^d$. Let ${\tilde L}^p (\mathbb R^d)$ be the space of ...
Akira's user avatar
  • 835
3 votes
1 answer
131 views

Spectrum of continuous functions as a semigroup

Let $X$ be a countable group (with the discrete topology) and let $C_b(X)$ be the ring of continuous bounded functions $X \to \mathbb{R}$. It is known that the maximal spectrum of $C_b(X)$, namely the ...
Serge the Toaster's user avatar
4 votes
1 answer
202 views

A problem on Demailly's proof of finiteness theorem of elliptic differential operator

I am reading Demailly's notes on pseudodifferential operators on manifolds. And I cannot understand a statement he had made when he tried to prove that the image of an elliptic differential operator ...
Skywalker's user avatar
  • 206
2 votes
1 answer
281 views

Global control of locally approximating polynomial in Stone-Weierstrass?

Let $X=\mathbb{R}$, and $\mathcal{A}:=\mathbb{R}[x]$ be the subalgebra (of $C(X)$) of univariate polynomials. Given $\varphi\in C_b(X)$ and $K\subset X$ compact, we know from Stone-Weierstrass that $$\...
fsp-b's user avatar
  • 463
3 votes
1 answer
337 views

Is there an operation in topology analogous to the operation of averaging over a compact subgroup in harmonic analysis?

Let me start with the following Illustration: Let $G$ be a compact group, and let $\pi:G\to H$ be its (surjective) continuous homomorphism onto a (compact) group $H$. So we can think that $H$ is the ...
Sergei Akbarov's user avatar
4 votes
0 answers
119 views

Is the range of a probability-valued random variable with the variation topology (almost) separable?

Let $X$ and $Y$ be uncountable Polish spaces, $\Delta(Y)$ be the space of Borel probability measures on $Y$ endowed with the Borel $\sigma$-algebra induced by the variation distance, and let $g:X\to \...
Michael Greinecker's user avatar

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