All Questions
Tagged with fa.functional-analysis metric-spaces
32 questions with no upvoted or accepted answers
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
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
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
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
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
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
$$
\left\{B\left(x_k,\frac1{n}\right)...
- 195
2
votes
0
answers
71
views
Perturbing the approximation property from the Lipschitz-free space to stay in the Wasserstein space
Let $(X,d,x)$ be a separable pointed metric space and let $\mathcal{F}(X)$ be its Arens-Eells (also called its Lipschitz-Free space; in the case where $X$ is Banach) space. We view the $1$-...
- 5,405
2
votes
0
answers
102
views
What is the relationship between barycenters in the Arens-Eells sense and barycenters in the optimal transport sense
Setup:
Let $X$ be a complete pointed metric space.
Let us briefly recall that the Wasserstein space $W_1(X)$ is identifiable with a subset of the Arens-Eells (or Lipschitz-Free) space $\operatorname{...
- 5,405
2
votes
0
answers
42
views
Generalized Hardy operator and Lorentz gamma spaces
I would like to find an inequality which would 'place' the generalized Hardy operator $\int_0^th(y)dy\int_y^tk^*(s)ds$ in between two Lorentz gamma spaces.
Any literature or ideas would be greatly ...
- 109
2
votes
0
answers
137
views
Conditions on the inequality with a gauge norm
Let $\Phi(x)=\int_0^x \phi(y)\,dy$, $x \in \mathbb{R}_+$, be an N-function, and let $u$ be locally inferable on $\mathbb{R}_+$. Consider the gauge norm
$$
\rho_{\Phi,u}(f)=\inf\{\lambda>0: \int_{\...
- 343
2
votes
0
answers
265
views
The contraction principle in quasi metric spaces
I am researching contractive mappings and I need the article of I. A. Bakhtin "The contraction principle in quasi metric spaces"(1989) or at least part where explanation is given for ...
2
votes
0
answers
210
views
A Riemannian metric on the plane such that the intersection of every two discs is a disc, again
Is there a Riemannian metric on $\mathbb{R}^2$ (or a $2$ dimensional manifold) such that the intersection of every two open discs is an open disc, again?
As linear version of this question we ask:
...
- 356
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
99
views
Equicontinuity of $\{f_{2n}\circ f_{2n-1}\}$
Let $(X,D)$ be a compact metric space and $\{f_n\}_{n\in\mathbb{N}}$ be a sequence of homeomorphisms of $(X,d)$. It is easy to see that if $\{f_n\}$ is uniformly convergent then $\{g_n\}$ defined by $...
2
votes
0
answers
126
views
Nearly injective Banach spaces
There was a problem about nearly injective metric spaces posed by Aronszajn and Panitchpakdi which I actually solved in the past but it still remains open (as long as I know) for the Banach spaces--so ...
- 7,500
1
vote
0
answers
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
1
vote
0
answers
449
views
Bound on covering number of Lipschitz functions – missing part in proofs of Kolmogorov et al
Given a metric space $(\mathcal{X},\rho)$ and $\mathcal{A}\subset\mathcal{X}$ totally bounded, i.e. $\mathcal{A}$ has a finite $\varepsilon$-covering for any $\varepsilon>0$. Consider $\...
- 11
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
1
vote
0
answers
53
views
Stability of Hajłasz-Sobolev class under post-composition
Informally: When is a Sobolev function, post-composed by a vector-valued function still Sobolev?
Assumptions/Setup
Let $(X,d_X,m_X)$ and $(Y,d_Y,m_Y)$ be complete and separable metric measure spaces; ...
- 5,405
1
vote
0
answers
70
views
Injectivity of post-composition operator
Let $X$, $Y_1,Y_2$, and $Z$ be separable metric spaces. Let $C(X,Y)$ be the topological space of continuous functions from $X$ to $Y$ equipped with its compact-open topologies. Fix a continuous ...
1
vote
0
answers
84
views
A Hölder version of the Johnson-Lindenstrauss Lemma on essentially bounded functions
Does there exist a Hölder (not necessarily linear) projection from $L^{\infty}(\mathbb{R}^d)$ to any finite-dimensional linear subspace? This is known when $L^{\infty}(\mathbb{R}^d)$ is replaced by a ...
- 5,405
1
vote
0
answers
105
views
Generalize characterization of upper semicontinous functions
Let $X$ be a metric space and denote $f:X \rightarrow \mathbb{R}.$
It is easy to show that the following two statements are equivalent:
$(1)$ For any real number $c$, we have $f^{-1}(-\infty,c)$ is ...
- 623
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
0
votes
0
answers
65
views
Let $E$ be Banach, $\mu_n\to\mu$ weakly on a locally compact $X$, and $f \in C_b(X, E)$. Does $\int f\mathrm d\mu_n\to\int f\mathrm d\mu$ in norm?
Let
$X$ be a metric space,
$(E, |\cdot|)$ a Banach space
$\mathcal M(X)$ the space of all finite signed Borel measures on $X$,
$\mathcal C_b(X)$ be the space of real-valued bounded continuous ...
- 835
0
votes
0
answers
45
views
Skorohod Space with $J_1$ topology homeomorphic to Frechet Space
Is the Skorohod space $D([0,T];\mathbb{R}^d)$ equipped with the $J_1$ topology homoeomorphic to a separable Fr\'{e}chet space. In particular, is it homeomorphic to $L_{\mu}^1(\mathcal{B}([0,1])$ ...
- 5,405