All Questions
72 questions
26
votes
2
answers
5k
views
Does Arzelà-Ascoli require choice?
Inspired by a recent Math.SE question entitled Where do we need the axiom of choice in Riemannian geometry?, I was thinking of the Arzelà--Ascoli theorem. Let's state a very simple version:
...
- 29.7k
15
votes
3
answers
1k
views
Version of Banach-Steinhaus theorem
I am wondering about the following version of the Banach-Steinhaus theorem.
Let $A$ be a closed convex subset contained in the unit ball of a Banach space $X$ and consider bounded operators $T_n \in \...
- 536
12
votes
2
answers
678
views
Non-sequential spaces in the wild
TLDR: What are examples of (function-)spaces that are not sequential? When does this matter?
As a simple analyst, I am most happy if I can just work with sequences all the time. In most situations ...
- 779
10
votes
1
answer
2k
views
Counting norms on an infinite dimensional vector space
It is known that whenever E is a finite dimensional real vector space, there is only one norm on E up to equivalence (actually one non discrete vector space topology).
Is it known what happens when E ...
- 101
9
votes
1
answer
831
views
Baire category theorem for uncountable unions
Any compact Hausdorff space $X$ is a Baire space:
if the set $X$ is a meager set (meaning a countable union of nowhere dense subsets,
also known as a set of first category),
then $X$ is empty.
I am ...
- 37.8k
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 $\sum a_n$ ...
- 3,104
8
votes
0
answers
110
views
Connected component optimization
For an open set $A\subset[0,1]^d$, denote the connected components of $A$ by $cc(A)$. Given a smooth symmetric function $f\colon[-1,1]^d\to\mathbb R$ with $f(0)>0$, I am interested in the ...
- 623
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 ...
- 21.2k
6
votes
3
answers
1k
views
functional subrings
I should recall the notion of maximal subring of a commutative unitary ring $R$.
Def: A commutative ring $S$ is called a maximal subring of $R$ if $S \subset R$ and if $T \subset R$ constitute a ...
- 1,788
5
votes
1
answer
805
views
Arzelà-Ascoli for $C_b(0,1)$? Or more generally, why is that continuous functions "live most naturally" on compact spaces?
I’m wondering if there is a version of Arzelà-Ascoli for continuous functions on not-necessarily compact metric/Hausdorff spaces $X$, i.e. a characterization of the compact subsets of $C_b(X)$ (under ...
- 831
5
votes
0
answers
99
views
What is a mild sufficient condition on $X$ such that $C(X, Y)$ is sequential?
Let $X$ be a topological space, $(Y, d)$ a metric space and $C(X, Y)$ the space of continuous maps with the topology of compact convergence.
Question: What is a minimal topological condition on $X$ ...
- 315
5
votes
0
answers
349
views
Tietze extension theorem for lower semi continuous functions
On the Tietze extension theorem, if instead of a continuous function "f" we use a lower semi continuous function on a closed subspace of a metric space, is the theorem correct? I mean, can we extend ...
- 179
4
votes
1
answer
224
views
Bounded growth of functions vs bounded growth of functions on countable sets
I am wondering if the boundedness of growth can be characterized by sequences. I am not sure if I use the term "growth" correctly, or use the correct tags for this question. Here is what I mean.
Let $...
- 5,529
4
votes
1
answer
228
views
Haar-null union of dense subsets
Let $\{X_i\}_{i \in \mathbb{R}-\{0\}}$ be a set of subsets of a separable infinite-dimensional Fréchet space $X$ and $I$ be uncountable. Moreover, suppose that
(Dense $G_{\delta}$) $X_i$ is a dense ...
- 63
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 ...
- 1,093
4
votes
1
answer
140
views
Whether a functional which preserves maximum for comonotone functions is monotone?
Let $X$ be a compactum (compact Hausdorff space). By $C(X,[0,1])$ we denote the space of continuous functions endowed with the sup-norm We also consider the natural lattice operations $\vee$ and $\...
- 61
4
votes
1
answer
121
views
Condition for existence of a continuous function realizing a partition
Let $\{U_i\}_{i=1}^{I}$ be a non-empty and finite collection of non-empty, disjoint, open, (and obviously bounded) subsets of $[0,1]^n$. Suppose also that $[0,1]^n=\cup_{i =1 }^{ I} \overline{U_i}$. ...
3
votes
2
answers
2k
views
Can every real function be approximated with a Riemann-integrable one with any precision required?
Is there some proof that Riemann-integrable functions are dense in the space of all real functions?
In a sense that for every real function $f$ and number $\varepsilon>0$, there is Riemann-...
- 117
3
votes
1
answer
684
views
Is the countably infinite product of locally convex topological vector spaces locally convex?
Let $(X,\tau)$ be a locally convex topological vector space and denote the product space
$$X^{\infty}=X\times X\times X\cdots:=\big\{x=(x_i)_{i\geq 1}:~ x_i\in X\big\}$$
If we endow $X^{\infty}$ ...
- 1,835
3
votes
1
answer
241
views
$\int_0^1 f(\sin(1/x)) \times g(\cos(1/x)) dx \leq \int_0^1 f(\sin(1/x)) dx \times \int_0^1 g(\cos(1/x))dx? $
I have noticed experimentally that the following question has a positive answer.
Is it true that for all even and convex functions $f$, $g$:
$$\int_0^1 f(\sin(1/x)) \times g(\cos(1/x)) dx \leq \int_0^...
- 4,074
3
votes
2
answers
676
views
Compact-open limit of continuous functions is continuous?
Let $X$ be a topological space and $Y$ a metric space.
A classical result states that compact-open topology on the space $C(X,Y)$ of continuous functions is the same as the topology of uniform ...
- 829
3
votes
1
answer
203
views
Regularize continuous functions with bounded variation
Is it true that :
$\forall f,g \in C([0,1],\mathbb R), \exists h \in C([0,1],[0,1])$ $f,g,h$ strictly increasing and $h([0,1])=[0,1]$ with $(f \circ h, g\circ h) \in C^{\infty}([0,1],\mathbb R)^2$?
- 4,074
3
votes
1
answer
352
views
Sequential separability on $C_p(X)$
Definition. Let $E$ be a topological space. Suppose that $E$ contains a sequence $\{x_n\}$ such that for every $x\in E$, there exists a subsequence $\{x_{n_k}\}$ of $\{x_n\}$ with $x=\lim x_{n_k}$. ...
- 4,058
3
votes
0
answers
187
views
Analogue of Kolmogorov/Arnold superposition for general manifolds?
Previously asked and bountied at MSE with slightly different language:
Given a topological space $\mathcal{X}$, let $$\mathsf{Cl_C}(\mathcal{X})=\bigcup_{n\in\mathbb{N}}C(\mathcal{X}^n,\mathcal{X})$$ ...
- 21.2k
2
votes
2
answers
154
views
Closure of $C([0,1]^2)$ via weak*-topology [closed]
Let $C([0,1]^2)$ denote the set of continuous functions on $[0,1]^2$. Let $L^1([0,1]^2)$ be the set of all Lebesgue integrable functions on $[0,1]^2$.
The dual space of $C([0,1]^2)$, denoted by $C^*([...
- 349
2
votes
1
answer
403
views
The set of Upper semi-continuous functions as a ring.
I should recall that the surgenfery topology on the real numbers is denoted by $\mathbb{R}_l$, and has the set
{$[a , b): a,b \in \mathbb{R} $} as it's base.
If $X$ is a topological space, an upper ...
- 1,788
2
votes
1
answer
336
views
Separability of $L^1$ in $L^2$ topology
In the space $L^1(0,1)$ take the topology generated by the $L^2$-balls
$$B^2_r(f)=\{g\in L^1(0,1):\; \|f-g\|_2<r\}.$$
Is $L^1(0,1)$ separable in this topology?
- 23
2
votes
1
answer
264
views
Is a continuous functional on continuous functions the restriction of a continuous functional on the space of all functions?
As sets, we can consider the space $C(\mathbf{R}^n;\mathbf{R}^k)$ - of all continuous functions from $\mathbf{R}^n$ to $\mathbf{R}^k$ - to be a subset of the product space $(\mathbf{R}^k)^{\mathbf{R}^...
- 1,179
2
votes
1
answer
281
views
Global control of locally approximating polynomial in Stone-Weierstrass?
Let $X=\mathbb{R}$, and $\mathcal{A}:=\mathbb{R}[x]$ be the subalgebra (of $C(X)$) of univariate polynomials.
Given $\varphi\in C_b(X)$ and $K\subset X$ compact, we know from Stone-Weierstrass that
$$\...
- 463
2
votes
2
answers
316
views
Properties of the topology of sequential convergence $\tau_\text{seq}$
Let $(X,\tau)$ be a Hausdorff space. Denote by $\tau_\text{seq}$ the topology on $X$ whose closed sets are the sequentially $\tau$-closed subsets of $X$. I have read that $\tau_\text{seq}$ has the ...
- 1,245
2
votes
1
answer
345
views
Function series of normal lower semi-continuous functions
For a real-valued $f$ on a topological space $X$, the upper limit of $f$ at $x\in X$ is
defined as follows:
$
f^{\ast }\left( x\right) =\inf \left\{ \sup \left\{ f\left( y\right) :y\in
U\right\} :U\in ...
- 1,367
2
votes
1
answer
265
views
characterization of normality by selection theorem
The Urysohn's extension theorem states that a space $X$ is normal iff every continuous function $f:A \rightarrow \mathbb{R}$, with $A$ a closed subset of $X$, can be extended to a continuous function $...
- 121
2
votes
1
answer
800
views
A question about Skorokhod metric
I have a question related to the Skorokhod distance.
Let $\Omega:=D([0,1],R)$ be the space of cadlag functions $x$ defined on $[0,1]$. Let $\Lambda$ be the collection of non-decreasing continuous ...
- 1,835
2
votes
1
answer
135
views
Skorokhod distance between $\omega, \omega\circ f_{\varepsilon}$ and $\omega, \omega\circ b_{\varepsilon}$
Let $\Omega:=D([0,1],R)$ be the space of cadlag functions $x$ defined on $[0,1]$. Let $\rho$ be the Skorokhod metric on $\Omega$, see e.g.
http://en.wikipedia.org/wiki/C%C3%A0dl%C3%A0g
Now define ...
- 1,835
2
votes
1
answer
128
views
Characterization of a subset of [0,1] $III$
I have a question related to the previous one.
Characterization of a subset of [0,1] $II$
Let $T\subseteq [0,1]$ be some subset closed under lower limit topology, i.e.
$t_n$ is said to converge to $...
- 1,835
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 \...
- 434
2
votes
0
answers
192
views
Generalize upper semicontinuous regularization using Borel Hierachy
Let $X$ be a metric space. Suppose a real-valued function $f:X\rightarrow \mathbb{R}$ is upper semicontinuous class $2$ if for all $c \in \mathbb{R},$ its preimage $f^{-1}(-\infty,c)$ is $F_{\sigma}$.
...
- 623
2
votes
0
answers
355
views
Existence of topology on the space of continuous functions
Let $C:=C([0,1],\mathbb{R})$ be the space of real-valued continuous functions defined on $[0,1]$. Could we find a topological vector space topology $\pi$ on $C$ such that the following two conditions ...
- 1,835
2
votes
0
answers
343
views
continuity with respect to weak-${\ast}$ topology
Let $V:=V([0,1],R)$ be the space of all cadlag functions defined on $[0,1]$ of bounded variation. Thus any element $v\in V$ determines a signed measure $\nu$ on $[0, 1]$ given by the formula $\nu([0, ...
- 1,835
2
votes
0
answers
76
views
question about a genralized Skorokhod topology
Let $D:=D([0,1], R)$ be the space of all cadlag functions defined on $[0,1]$. Now we have the known Skorokhod topology defined by: $\forall f, g\in D$
$$\rho(f,g):=\inf_{\lambda\in\Lambda}\Big\{\max\...
- 1,835
1
vote
1
answer
162
views
Is there a uniformly continuous injective image of $(0,1)\setminus\Bbb Q$ in the Cantor space?
It seems too good to be possible, but:
Is there a uniformly continuous injective image of $(0,1)\setminus\Bbb Q$ in the Cantor space?
Here, the Cantor space $\{0,1\}^{\Bbb N}$ is equipped with the ...
- 3,104
1
vote
1
answer
263
views
Does global boundedness ruin Stone-Weierstrass denseness?
Let $X$ be any topological space and denote by $\tau_X$ the topology on $C_b(X;\mathbb{R})$ that is induced by the family of seminorms $(\|\cdot\|_\psi\mid\psi\in B_0(X))$ with $\|f\|_\psi:=\sup_{x\in ...
- 463
1
vote
1
answer
118
views
Almost periodic function and closed spaces
We denote $X_{T}$ the vector space of all $T$-periodic function with zero mean in $L^2$ ( we know that $X_{T}$ is spawn by $(e^{2i\pi nt/T})$). Let be $$X=X_{2\pi}+X_{3\pi}.$$
I think that $X_{2\pi}+...
- 75
1
vote
1
answer
162
views
Does there exist a class of real-valued upper semicontinuos functions on $X$ such that $\mathcal{F}$ is countable?
Ian Morris quoted the following:
For any upper semi-continuous function $f \colon X \to [-\infty,+\infty)$ defined on a nonempty topological space $X$ there exists a nonempty set $\mathcal{F}\...
- 623
1
vote
0
answers
178
views
Density of subspace with nonlocal/Wentzell boundary condition
Given the space $F$ defined by:
$$F=\left\{f\in C^2(\mathbb{R}_+^2;\mathbb{R}):f(x,0)=\int_\mathbb{R} f(z,x)g(z)dz, x>0\right\},$$
I want to prove that the subspace $E$ of $F$ defined by $E=\...
- 111
1
vote
0
answers
260
views
Generating the sigma algebras on the set of probability measures
I was wondering if somebody could help me see/provide a reference to the following fact: Let $X$ be a metrizable set, $\mathcal{F}$ the corresponding Borel sigma-algebra on $X$, and $\triangle\left(X,\...
- 11
1
vote
0
answers
525
views
Separability of the space $C(C[0, 1], \mathbb{R})$
Let $E=C([0, 1])$ be the space of all real-valued continuous functions on $[0, 1]$, equipped with the uniform norm. $C(E)$ stand for the continuous real-valued functions on $E$.
I am wondering that ...
- 199
1
vote
0
answers
331
views
Relationship between weak Lp and strong Lq topologies for q<p
Specificaly:
Does convergence in $L^{\frac{1}{2}}$ imply weak $L^2$ convergence?
Having a limit in $L^{\frac{1}{2}}$ topology and a limit in weak $L^2$ topology whether these are always equal? If not,...
- 203
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 $[0,1]^{\Bbb N}$ ...
- 3,104
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 ...
- 349