All Questions
448 questions
1
vote
1
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263
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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
5
votes
1
answer
199
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Is the unit ball of $B(H)$ a Baire space (with the SOT)?
Let $H$ be a Hilbert space, and let $B(H)$ be the set of bounded linear operators $t \colon H \to H$. Recall that we say $t_i \to t$ in the strong operator topology if $t_i \xi \to t \xi$ for every $\...
- 733
0
votes
0
answers
77
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Completeness of a normed space
We consider the set $\mathcal{PC}([-r,0],X)$
$$\mathcal{PC}([-r,0],X):=\{\varphi:[-r, 0] \rightarrow X: \varphi \text{ is continuous everywhere except
for a finite number
of points } t_* \text{ ...
- 39
5
votes
1
answer
165
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Algebraic solutions of polynomial ODEs
Given a polynomial ODE in $n$-dimensions of maximal degree $d$
$$
\dot{x}_j=f_j(x)=\sum_{i_{1},\dots,i_{n}=1}^{d}a_{i_{1},\dots,i_{n}}^{j}x_{1}^{i_{1}}\dots x_{n}^{i_{n}} \quad \forall j=1,...,n
...
- 247
17
votes
1
answer
569
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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 ...
- 775
6
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1
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500
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A characterization of metric spaces, isometric to subspaces of Euclidean spaces
I am looking for the reference to the following (surely known) characterization of metric spaces that embed into $\mathbb R^n$:
Theorem. Let $n$ be positive integer number. A metric space $X$ is ...
- 41.9k
7
votes
1
answer
331
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A metric characterization of Hilbert spaces
In the Wikipedia paper on Hadamard spaces, it is written that every flat Hadamard space is isometric to a closed convex subset of a Hilbert space. Looking through references provided by this Wikipedia ...
- 41.9k
4
votes
0
answers
152
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Maximally fine topologies on $B(H)$ making the unit ball compact
Let $H$ be a Hilbert space, and $B(H)$ its algebra of bounded operators. One of the reasons the Ultraweak topology is (in a way) more useful than the weak operator topology is that the Ultraweak ...
- 190
3
votes
1
answer
451
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Topological vector spaces in direct sum
A year ago, I asked this question here at Mathematics Stackexchange, but no one there managed to answer it. So I am elevating it to MathOverflow.
This question had emerged as an offshoot of a bigger ...
- 603
2
votes
1
answer
192
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Open sets in the space of signed measures equipped with the Kantorovich–Rubinshtein norm
Let $X$ be a compact metric space and $\mathcal{M}(X)$ be the space of variational-bounded, signed Borel measures equipped with the Kantorovich–Rubinshtein norm, cf. [Section 8.3, 1]:
$$||\mu||_0:= \...
- 113
11
votes
2
answers
721
views
Existence of an open convex set
Let $T$ be a normed vector space, $K\subseteq T$ compact and convex and $O\subseteq K$ convex and open in $K$ (i.e. open w.r.t. the subspace topology of $K$ inherited by $T$).
Can we find an open set $...
- 113
0
votes
0
answers
131
views
Cyclic group action and finite invariant set
Let $(X, d)$ be a compact metric space and $G$ a discrete group acting on $X$ such that, for each $g\in G$, the mapping $x\mapsto g\cdot x$ defines a homeomorphism on $X$
Is it true that the ...
- 307
8
votes
4
answers
681
views
Uniform density of Lipschitz maps is space of continuous function — for general metric spaces
Let $X$ and $Y$ be metric space, $X$ be compact, $C(X,Y)$ denote the set of continuous functions from $X$ to $Y$ with uniform convergence on compacts topology, and $\operatorname{Lip}(X,Y)$ denote the ...
- 364
3
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0
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175
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Any reference on Jensen inequality for measurable convex functions on a Hausdorff space?
I asked this question on math.stackexchange and I was suggested that asking it may be more appropriate. This is part of my research which tries to extend some of Choquet's theory to some non-compact ...
- 204
3
votes
1
answer
355
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Extremely disconnected or extremally disconnected?
In the context of Banach space theory, what is the correct terminology: extremally disconnected or extremely disconnected. Looking through the internet I have met using both extremely and extremally ...
- 41.9k
2
votes
1
answer
183
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On the equality $\{f\in C(X), f|_A=0\}+\{f\in C(X), f|_B=0\}=\{f\in C(X), f|_{A\cap B}=0\}$
Let $X$ be a Tychonoff space, let $A,B\subset X$ be closed. Let $J_A$ be the set of all continuous on $X$ real-valued functions which vanish on $A$.
For which $X$'s is it true that $J_A+J_B=J_{A\cap ...
- 5,529
2
votes
0
answers
52
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Can we decompose an increasing net of functions into two increasing nets with prescribed supports?
Let $K$ be a compact Hausdorff space and let $U,V\subset K$ be open. Let $\left(f_{i}\right)_{i\in I}$ be an increasing net of continuous non-negative functions such that $f_{i}\le 1$ and $f_{i}$ ...
- 5,529
1
vote
0
answers
99
views
Density of Lipschitz functions in Bochner space with bounded support
Let $X$ and $Y$ be separable and reflexive Banach spaces with Schauder bases. Let $\mu$ be a non-zero finite Borel measure on $X$ and let $L^p(X,Y;\mu)$ denote the (Boehner) space of strongly p-...
- 21
0
votes
1
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290
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Tensor product is complete?
Let $(V,\|\cdot\|_V)$ and $(W,\|\cdot\|_W)$ be Banach spaces and let the norm $\|\cdot\|_{V\otimes W}$ on the tensor product space $V\otimes W$ be admissible in the following sense: for $v\in V, w\in ...
- 177
4
votes
1
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140
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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
0
votes
1
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243
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Condition for set of the type $\{(a,b)|a \in A, \ b = f(a)\}$ to have empty interior if $A$ has empty interior [closed]
Let us consider $$\mathcal X = \{(a,b)|a \in A, \ b = f(a)\}, $$ where $A \subset L^1(\mathbb R)$ has empty interior and $f:L^1 \to L^1$ is a bijective map. Does $\mathcal X$ also have empty interior? ...
- 839
13
votes
2
answers
2k
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When can we divide continuous functions?
Let $X$ be a compact Hausdorff topological space such that for every continuous $f,g:X\to\mathbb{R}$ with $0\le f\le g$ there is a continuous $h:X\to\mathbb{R}$ such that $f=gh$.
What can be said ...
- 5,529
8
votes
1
answer
1k
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Compactness of the unit ball of a Banach space for topologies finer than the weak* topology
Let $(\mathcal{X} , \|\cdot \|_\mathcal{X})$ be a Banach space and $\mathcal{X}'$ its topological dual. We denote by $\| \cdot \|_{\mathcal{X}'}$ the dual norm and define also the topological dual $\...
- 2,306
1
vote
0
answers
54
views
Closed linear span of compact open subsets of a spectral space
Let $X$ be a spectral space and $KO(X)$ be the set of all compact open subsets of $X$. Identify $KO(X)$ with $\{1_D:D\in KO(X)\}$, where $1_D(u) = 1$ if $u\in D$ and $1_D(u) = 0$ if $u\notin D$.
...
- 2,605
3
votes
0
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126
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A path with zero increments and positive area
I am studying rough paths from the 2007 St Flour lecture notes and I came across the example at the end of chapter one of the sequence of paths $X(n):[0,2\pi]\to \mathbb R^2$ given by $X_t(n) = \frac{...
- 177
0
votes
0
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98
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Reference request: subspace-based generalisation of weak* convergence
Let $V$ be a normed space and $(V_j)_{j\in [0,1]}$ be a family of linear subspaces of $V$ with $V_1$ non-trivial and such that $V_1\subsetneq V_j\subseteq V_i$ whenever $i\leq j$. We write $W:=V'$ for ...
- 463
1
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0
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97
views
Are Hölder functions between Banach spaces residual in the compact-open topology?
Let $X$ and $Y$ be Banach spaces and let $C(X,Y)$ be the set of continuous functions from $X$ to $Y$ equipped with the topology of uniform convergence on compact sets (i.e. the compact-open topology). ...
- 5,405
2
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0
answers
75
views
Dual space induced by a finer topology
Let $\|\cdot\|_1$ and $\|\cdot\|_2$ be two seminorms on a space $E$ such that $\|\cdot\|_2\geq\|\cdot\|_1$. Let further $E_i:=(E,\|\cdot\|_i)$ and
$$C_b(E_i):=\{f : E\rightarrow\mathbb{R}\mid f \ \...
- 463
3
votes
0
answers
153
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Is it possible to reconstruct the universally measurable sets in X from the $C^*$-algebra $C(X)^{**}$?
This continues my question of two months ago. Let $X$ be a compact Hausdorff topological space. We consider the $C^*$-algebra $C(X)$ of continuous functions on $X$, its dual space $C(X)^{*}=M(X)$ of ...
- 7,426
2
votes
2
answers
252
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When is the Minkowski sum of weighted compact sets $w_1 B_1 + w_2 B_2 + \ldots$ (with $w \in L^1$) closed?
Let $B_1,B_2,\ldots,$ be compact subsets of $\mathbb R^d$ and $w_1,w_2,\ldots$ be nonnegative numbers summing to $1$. Consider the set
$$
A := w_1 B_1 + w_2 B_2 + \ldots = \left\{\sum_{n=1}^\infty w_n ...
- 6,853
1
vote
3
answers
345
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Under what general conditions is the set $S := \left\{\int_{X}v(x)\pi(x)\,\mathrm{d}P(x) \mid \pi: X \to A\right\}$ closed?
Let $X$ be a compact subset of $\mathbb R^n$ and let $A$ be a compact subset of $\mathbb R^k$. Let $P$ be a probability distribution on $X$ and $v$ be a $P$-measurable function from $X$ to $\mathbb R^{...
- 6,853
1
vote
0
answers
112
views
What is the topological characteristic of a separable metric space $X$ such that $|kX\setminus X|=\frak{c}$ for any completion $kX$ of $X$?
What is the topological characteristic of a separable metric space $X$ such that $|kX\setminus X|=\frak{c}$ for any completion $kX$ of $X$?
- 1,101
7
votes
0
answers
2k
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Algebraizing topology and analysis via condensed mathematics
I asked this question on Mathematics Stackexchange, but one of the users suggested that I ask this question at MathOverflow.
I've just come across a Twitter thread by Laurent Fargues explaining a work ...
- 79
3
votes
0
answers
233
views
Is it possible to reconstruct the compact space $X$ from the space of measures $M(X)$?
Let $X$ be a compact Hausdorff topological space and $C(X)$ the Banach algebra of continuous functions $u:X\to\mathbb C$ (with the usual $\sup$-norm). It is well-known that the structure of Banach ...
- 7,426
5
votes
1
answer
420
views
Ring of continuous functions is a Jacobson ring
Let $X$ be an infinite discrete topological space. Is $$C_b(X)=\{ f \colon X \to \mathbb{R} \text{ bounded }\}$$ a Jacobson ring ?
13
votes
2
answers
915
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Topological vector spaces (reference request)
In his book Topological Function Spaces Arhangel'skii says that "it is well known that every nontrivial locally convex linear topological space $X$ is homeomorphic to a space of the form $Y \...
- 674
15
votes
2
answers
931
views
Distinguishing topologically weak topologies of Banach spaces
Are the weak topologies of $\ell_1$ and $L_1$ homeomorphic?
Strangely may it sound, the question seeks contrasts between norm and weak topologies of Banach spaces from the non-linear point of view. ...
- 11.3k
3
votes
2
answers
2k
views
Can every real function be approximated with a Riemann-integrable one with any precision required?
Is there some proof that Riemann-integrable functions are dense in the space of all real functions?
In a sense that for every real function $f$ and number $\varepsilon>0$, there is Riemann-...
- 117
5
votes
1
answer
521
views
Properties of $C_B(X)$ equipped with the strict topology
Let $X$ be a Polish space. $C_B(X)$ is the space of bounded continuous functions $X\to\mathbb{R}$ equipped with the strict topology, which is the finest locally convex topology that agrees with the ...
- 353
0
votes
0
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94
views
Is the space of affine continuous functions a Baire space
Let $\Omega$ be a compact convex set in q linear normed space. Let $A(\Omega)$ be the space of affine continuous real-valued functions. My question is whether the space $A(\Omega)$ is a Baire space? ...
- 675
3
votes
1
answer
241
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$\int_0^1 f(\sin(1/x)) \times g(\cos(1/x)) dx \leq \int_0^1 f(\sin(1/x)) dx \times \int_0^1 g(\cos(1/x))dx? $
I have noticed experimentally that the following question has a positive answer.
Is it true that for all even and convex functions $f$, $g$:
$$\int_0^1 f(\sin(1/x)) \times g(\cos(1/x)) dx \leq \int_0^...
- 4,074
2
votes
1
answer
161
views
Existence of a Borel measurable function
Let $X$ be a compact metric space and $Y\subset X$ be a compact set. Assume that $f_1, f_2: Y \to \mathbb{P}\mathbb{R}^2$ are continuous functions. Let $N \subset \mathbb{P}\mathbb{R}^2$ be a ...
- 133
3
votes
2
answers
306
views
Smoothing a map $f:X\to \mathbb{R}$ while fixing it over a closed $C\subset X$
$\newcommand{\R}{\mathbb{R}}$I have a map $f\in C^0(X,\mathbb{R})$, where $X$ is a compact and Hausdorff topological space, which is a manifold outside of a compact subset $K\subset X$.
I would like ...
- 2,533
3
votes
1
answer
353
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
1
vote
1
answer
172
views
A question about pushforward measures and Peano spaces
Specifically my question is the following: Let $P$ be a Peano space. If $(P,\sigma,\mu)$ and $(P,\sigma,\nu)$ are both nonatomic probability measures, does there exist a continuous function $f:P\to P$ ...
- 33
1
vote
1
answer
232
views
An approximation property in a separable topological vector space
Let $X$ be a topological vector space.
Let us say that $X$ enjoys sequential separablity if there exists a sequence $\{x_n\}$ in $X$ such that for every $x\in X$ there exists a subsequence of $\{...
- 4,058
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 ...
- 833
2
votes
0
answers
198
views
A generalisation of closed and bounded subsets of non-Archimedean fields to topological spaces
The definition of compactness in topological spaces generalises the notion of a subset of $\mathbb{R}^n$ being closed and bounded, as expressed by the Heine-Borel Theorem.
In finite-dimensional vector ...
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 $$\mathsf{Cl_C}(\mathcal{X})=\bigcup_{n\in\mathbb{N}}C(\mathcal{X}^n,\mathcal{X})$$ ...
- 20.5k
5
votes
2
answers
359
views
Product of inductive limit topologies on $C_c(X)\times C_c(X)$
I have a stupid question about a topology on $C_c(X)$. Here $X$ is locally compact Hausdorff. Can assume $\sigma$-compact if it helps.
Definition (topology on $C_c(X)$): For each compact $K \subset X$,...
- 675