All Questions
Tagged with fa.functional-analysis metric-spaces
78 questions
15
votes
2
answers
2k
views
In infinite dimensions, is it possible that convergence of distances to a sequence always implies convergence of that sequence?
This is a cross-posted on MSE here.
Let $(X,d)$ be a metric space. Say that $x_n\in X$ is a P-sequence if $\lim_{n\rightarrow\infty}d(x_n,y)$ converges for every $y\in X.$ Say that $(X,d)$ is P-...
- 1,225
13
votes
1
answer
3k
views
metric on the space of real analytic functions
Hello,
this question may be simple but I couldn't find a reference.
Let $E$,$F$ be real Banach spaces and $\Omega\subset E$ be a bounded domain and let $C_b^{\omega}(\Omega,F)$ be the vector space of ...
- 223
13
votes
1
answer
4k
views
Modulus of Continuity
I originally posted this question on math.stackexchange (https://math.stackexchange.com/questions/83182/modulus-of-continuity-take-2), but it's been a few days and I haven't received any correct ...
- 29.2k
13
votes
1
answer
3k
views
Does this metric have an official name? Lévy metric? Ky Fan metric?
Let $X$ and $Y$ be random variables taking values in a separable metric space $(S,d)$. The metric I have in mind is
$$\rho(X,Y) = \mathbb{E}[\min\{d(X,Y),1\}]$$
if $X$ and $Y$ take values in the a ...
- 6,287
13
votes
0
answers
818
views
Covering number estimates for Hölder balls
Let $\alpha \in (0,1]$, $r>0$ and $L>0$, and positive intwgers $n$ and $m$. The Arzela-Ascoli Theorem guarantees that the set $X(\alpha,L,r)$ of $f:[-1,1]^n\rightarrow [-r,r]^m$ with $\alpha$-...
- 5,405
12
votes
5
answers
1k
views
Examples of metric spaces with measurable midpoints
Given a (separable complete) metric space $X=(X,d)$, let us say $X$ has the measurable (resp. continuous) midpoint property if there exists a measurable (resp. continuous) mapping $m:X \times X \to X$ ...
- 6,853
12
votes
1
answer
575
views
Is $\ell_p$ $(1<p<\infty)$ finitely isometrically distortable?
Let $Y$ be a Banach space isomorphic to $\ell_p$, $1<p<\infty$. Is it true that any finite subset of $\ell_p$ is isometric to some finite subset of $Y$?
It seems to me that it is an interesting ...
- 4,878
9
votes
4
answers
4k
views
Is the space of Radon measures a Polish space or at least separable?
Background: I work on a SPDE problem where in order to apply Prokhorov's theorem I need that some measure space is Polish space. And additionaly it would be good if that space is Banach space. Earlier ...
- 657
9
votes
1
answer
311
views
Continuously varying norms
Let $V$ be a finite-dimensional real vector space with its Euclidean topology. Then all norms on $V$ are equivalent and consequently given two norms $\lVert-\rVert$, $\lVert-\rVert'$, the number
$$
d =...
- 1,153
8
votes
4
answers
681
views
Uniform density of Lipschitz maps is space of continuous function — for general metric spaces
Let $X$ and $Y$ be metric space, $X$ be compact, $C(X,Y)$ denote the set of continuous functions from $X$ to $Y$ with uniform convergence on compacts topology, and $\operatorname{Lip}(X,Y)$ denote the ...
- 364
8
votes
2
answers
689
views
Topological characterization of injective metric spaces
Let $\ (X\ d)\ \,(Y\ \delta)\ $ be arbitrary metric spaces. A function $\ f:X\rightarrow Y\ $ is called a metric map (with respect to the given metrics $\ d\ \delta$) $\ \Leftarrow:\Rightarrow\ \...
- 7,500
7
votes
1
answer
773
views
Equivalent metrics on Fréchet spaces and Lipschitz maps
Lipschitz maps are defined over metric space as maps $f:(X,d_X) \to (Y,d_Y)$ such that
$$ d\left( f(x),f(x^\prime) \right)_Y \le k d(x,x^\prime)_X \ \forall x,x^\prime \in X, $$
where $k$ is a ...
- 399
7
votes
0
answers
150
views
The space of analytic associative operations
This question is a follow-up to this old one of mine.
Let $\mathcal{A}$ be the set of functions $\star:\mathbb{R}^2\rightarrow\mathbb{R}$ which are associative and $C^\omega$ (real analytic entire) in ...
- 20.9k
6
votes
2
answers
424
views
Lipschitz mappings, covering dimension
Is there a compact metric space $X$ of covering dimension $2$ without a Lipschitz surjection on $[0,1]^2$?
For a space $X$ with Hausdorff dimension greater than $2$, we have a negative answer (see ...
- 97
6
votes
1
answer
348
views
Reference: Hajlasz-Sobolev Spaces with Values in a Metric Space
Let $(X,d,\mu)$ be a separable metric measure space on which every ball has positive but finite measure.
I've come across the definition of a homogeneous Fractional Hajlasz-Sobolev spaces $M^{s,p}(...
- 5,405
6
votes
0
answers
182
views
Factorization of metric space-valued maps through vector-valued Sobolev spaces
Let $(X,d,m)$ and $(Y,\rho,n)$ be metric measure spaces and let $f:X\rightarrow Y$ be a Borel-measurable function for which there is some $y_0$ and some $p\geq 0$ such that
$$
\int_{x\in X}\,d(y_0,f(x)...
- 5,405
5
votes
4
answers
4k
views
Non-separable Banach space
The vector space $C_b(\mathbb R)$ of bounded continuous functions on $\mathbb R$ is non-separable: it is possible to produce a direct proof of this fact, mimicking the standard proof for the non-...
- 16.2k
5
votes
1
answer
148
views
Continuity of central point operation
Stanisław Mazur and Stanisław Ulam, in their joint paper, characterized the mid-point $\ \frac{a+b}2\ $ in a Banach space in pure metric terms (without algebra). This allowed them to show that any two ...
- 7,500
5
votes
1
answer
483
views
Can you always extend an isometry of a subset of a Hilbert Space to the whole space?
I remember that I read somewhere that the following theorem is true:
Let $A\subseteq H$ be a subset of a real Hilbert space $H$ and let $f : A \to A$ be a distance-preserving bijection, i.e. a ...
- 609
5
votes
2
answers
642
views
Is the Hausdorff metric on sub-$\sigma$-fields separable?
Let $(X,\mu,\mathcal{F})$ be a probability space. The paper Equiconvergence of Martingales by Edward Boylan introduced a pseudometric on sub-$\sigma$-fields (sub-$\sigma$-algebras) of $\mathcal{F}$ ...
- 6,287
4
votes
2
answers
191
views
Reference request: "Tangent relation" in metric spaces
Let $X,Y$ be metric spaces. Let $f,g : X \to Y$ be two maps and $x_0 \in X$. Let us say that $f$ and $g$ are tangent at $x_0$ if the following condition is satisfied: For every $\epsilon > 0$ there ...
- 63.1k
4
votes
2
answers
374
views
Vague convergence: confusion about the regularity of a signed Radon measure and that of its variation
I'm reading a proof of below theorem from this paper.
Theorem A.3. Let $\Omega$ be a locally compact normal Hausdorff space. Let $\left\{\mu_n\right\} \cup\{\mu\} \subset \mathcal{M}(\Omega)$ and ...
- 657
4
votes
1
answer
104
views
Generalization of a bounded variation
Let $(X, d)$ be a metric space. We will say that $\gamma \colon [a,b] \to X$ is of bounded variation, if
\begin{equation}
V(\gamma) = \sup_{a=t_0 < \cdots < t_n < b} \sum_{i=1}^n d( \gamma(...
- 820
4
votes
0
answers
197
views
Compactness of the unit ball in the space of Radon measures w.r.t. the Kantorovich-Rubinstein norm
This question was posted previously but has not attracted any responses so I am repharising it in a slightly different language hoping to reach a wider community
Let $(X,d)$ be a pointed metric space ...
- 437
4
votes
0
answers
147
views
Continuous extension preserving modulus of continuity
Let $X$ be a (non-empty) compact subset of $D(0,M):=\left\{x\in \mathbb{R}^n:\, \|x\|\leq M\right\}$, and let $f:X\rightarrow Y$ be uniformly continuous; for some metric space $Y$. Are there any ...
4
votes
0
answers
159
views
Is there a name for this geometric property of metric spaces?
My research has lead me to metric spaces $(M, \rho)$ which have the following geometric property:
Suppose $x, y \in M$ and $r, s > 0$ such that
$(x, r) \neq (y, s)$,
$B[y; s] \subseteq B[x; r]$,
$...
- 191
4
votes
0
answers
99
views
Fractional Hajłasz-Besov-like similar to the Korevaar-Schoen-Sobolev spaces?
Suppose that $(X,\mu,d)$ and $(Y,\nu,\rho)$ are doubling metric measure spaces. Fix $\alpha>0$ and define the space, analogously to this paper, as the collection of all measurable functions $f:X\...
- 5,405
3
votes
1
answer
806
views
Under what conditions can we put a complete norm on a linear subspace of a separable Banach space?
Question 1 Let $X$ a separable Banach Space and $Y\subset X$ linear subspace. When can we put a norm on $Y$ in such a way so that $Y$ is a Banach space?
Clearly if $Y$ is closed in the norm topology ...
- 243
3
votes
1
answer
161
views
Equivalent definition for Skorokhod metric
I have a question about the Skorokod distance on the space $\mathcal{D}([0,1],\mathbb{R})$:
$$
d(X,Y):= \inf_{\lambda \in \Lambda}\left( \sup_{t\in [0,1]}|t-\lambda(t)|\vee \sup_{t\in [0,1]}|X(t)-Y(\...
- 177
3
votes
1
answer
182
views
Lipschitz-free space of countable uniformly discrete metric space
I assume here that the reader is familiar with the concept of Lipschitz-free space $\mathcal{F}(X)$ of a metric space $X$. I will follow the definition of $\mathcal{F}(X)$ as the completion of the ...
3
votes
1
answer
148
views
Is this sum of nonexpansive maps itself nonexpansive?
For Hilbert spaces $\mathcal{H}_X$, $\mathcal{H}_Y$ and $\mathcal{K}$, consider a linear map $f\colon \mathcal{K} \oplus \mathcal{H}_X \to \mathcal{K} \oplus \mathcal{H}_Y$ that is given as a matrix
$...
3
votes
1
answer
233
views
Show identity for a norm on Fréchet differentiable functions on a Banach space
Let $E$ be a $\mathbb R$-Banach space, $v:E\to(0,\infty)$ be continuous with $$\inf_{x\in E}v(x)>0\tag1,$$ $r\in(0,1]$ and$^1$ $$\rho(x,y):=\inf_{\substack{c\:\in\:C^1([0,\:1],\:E)\\ c(0)=x\\ c(1)=...
- 167
3
votes
0
answers
132
views
Is the Schwartz space a tame Frechet space?
I ran into the following definition of tame Frechet spaces and Nash-Moser therem.
It says that the space of smooth functions on a compact manifold is tame Frechet.
However, I wonder if
The Schwartz ...
- 3,477
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 ...
3
votes
0
answers
171
views
Covering number $C^k$-balls in $C(\mathbb{R}^n)$
Fix a positive integer $n$ and and an non-negative integer $k$. The Arzela-Ascoli theorem guarantees that for a given positive integer $k$ and a given $L>0$ the set
$$
Ball_{C^{k,1}([0,1]^n)}(0,L)
...
- 5,405
3
votes
0
answers
89
views
Reference request: Projection operators in metric spaces
Given a metric space $(X,d)$ and a subset $S\subset X$, the projection $P_S$ onto $S$ is well-defined as a set valued function. I am interested in learning more about properties of these projections ...
- 710
3
votes
0
answers
487
views
Homeomorphism between $L^p$-spaces on metric spaces and $L^p$-spaces on Euclidean space
Setup:
Fix $p \in [1,\infty)$.
Let $(X,d_X,x_0)$ and $(Y,d_Y,y_0)$ be complete pointed metric spaces and $\mu$ be Borel. Let $E^n,E^D$ be Euclidean spaces of respetive dimensions $n$ and $D$ and ...
- 5,405
2
votes
1
answer
236
views
A sensible topology on the space of continuous linear maps between Fréchet spaces
Let $V_1$ and $V_2$ be Fréchet spaces. Let $\{ \lVert \cdot \rVert_{1,n} \}_{n \in \mathbb{N}}$ be a family of seminorms for $V_1$ and similarly $\{ \lVert \cdot \rVert_{2,n} \}_{n \in \mathbb{N}}$ ...
- 3,477
2
votes
3
answers
299
views
Banach-like analysis on metric spaces
Some time ago, I was thinking about whether it would be possible to generalize some results from functional analysis on Banach spaces to some metric spaces. Specifically, I wondered whether if one ...
- 820
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 ...
- 5,529
2
votes
1
answer
336
views
Complex Borel measures: does $\mu_n \to \mu$ weakly imply $|\mu|(\Theta) \le \liminf_n |\mu_n|(\Theta)$ for every open subset $\Theta$?
Let
$\Omega$ be a metric space,
$C_b(\Omega)$ the space of all real-valued bounded continuous functions on $\Omega$, and
$\mathcal{M}(\Omega)$ the space of all finite signed Borel measures on $\Omega$...
- 657
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: \...
- 55
2
votes
1
answer
87
views
Connected metric spaces without connected separable subspaces
My question is the following:
Does there exist a connected metric space $\ X,\ $ where $\ |X|>1,\ $ which contains no separable connected subspace $\ Y\ $ with $\ |Y|>1\ $?
- 7,500
2
votes
1
answer
89
views
Upper bound on the Levy-Prokhorov distance between the distributions of continuous Gaussian processes in terms of their covariances
Denote by $d$ the supremum metric on the space $C[0,T]$ of continuous real-valued functions on $[0,T]$:
$$
d(f,g) = \sup_{t \in [0,T]} |f(t)-g(t)|.
$$
Let $\rho$ be the Levy-Prokhorov metric on the ...
- 177
2
votes
1
answer
1k
views
Proving that family of sets has non-empty intersection
Let's say I have an object which can be viewed as family of sets $\mathfrak{S} \subseteq 2^S$, and I want to prove that its intersection is non-empty. What is known already:
$S$ is set of measurable ...
- 105
2
votes
1
answer
110
views
Lipschitz maps with Hölder inverse preserve the doubling property
Let $K$ be a compact doubling metric space, $X$ be a metric space and $f:K\rightarrow X$ be Lipschitz with $\alpha$-Hölder inverse, where $0<\alpha<1$. Does $f(K)$ need to be doubling?
- 5,405
2
votes
1
answer
223
views
Is there a theory of partially-defined metric spaces?
Is there a theory of metric spaces in which the distance between a given pair of points need not be defined?
I'm aware that there is a theory of partial metric spaces, but these deal with a different ...
- 3,065
2
votes
0
answers
93
views
Finite approximations to the Kuratowski/Fréchet embedding
Let $(X,d)$ be a compact doubling metric space with doubling constant $C>0$. Let $\{\mathbb{X}_n\}_{n=0}^{\infty}$ be a sequences of finite subsets of $X$ with
$$
\left\{B\left(x_k,\frac1{n}\right)...
- 195
2
votes
0
answers
71
views
Perturbing the approximation property from the Lipschitz-free space to stay in the Wasserstein space
Let $(X,d,x)$ be a separable pointed metric space and let $\mathcal{F}(X)$ be its Arens-Eells (also called its Lipschitz-Free space; in the case where $X$ is Banach) space. We view the $1$-...
- 5,405
2
votes
0
answers
102
views
What is the relationship between barycenters in the Arens-Eells sense and barycenters in the optimal transport sense
Setup:
Let $X$ be a complete pointed metric space.
Let us briefly recall that the Wasserstein space $W_1(X)$ is identifiable with a subset of the Arens-Eells (or Lipschitz-Free) space $\operatorname{...
- 5,405