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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 $\alpha$-...
ABIM's user avatar
  • 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 $m:X \times X \to X$ ...
dohmatob's user avatar
  • 6,853
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 ...
Math_Newbie's user avatar
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\ \...
Włodzimierz Holsztyński'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
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-...
Bazin's user avatar
  • 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 ...
Włodzimierz Holsztyński's user avatar
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
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 ...
ABIM's user avatar
  • 5,405
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
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
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 ...
Doktor Diagoras's user avatar
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 $F_{\sigma}$. ...
Idonknow's user avatar
  • 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 $...
Ali Barzanouni's user avatar
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). ...
ABIM's user avatar
  • 5,405
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 $X$?
Alexander Osipov's user avatar
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 ...
SetValued_Michael's user avatar
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}\...
Idonknow's user avatar
  • 623
0 votes
1 answer
223 views

Dense $G_{\delta}$ set with $\sigma$-porous complement is cofinite?

Let $X$ be a separable Banach space and $D\subseteq X$ be a proper, connected, and dense $G_{\delta}$ subset of $X$, $X-D$ is $\sigma$-porous. Then is $X-D$ contained in a finite-dimensional ...
ABIM's user avatar
  • 5,405
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 ...
Sanae Kochiya's user avatar