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0 votes
1 answer
100 views

Embeddings of pseudo metric spaces into seminormed Spaces

There is a theorem stating that every metric space embeds isometrically into $\ell _{\infty}$. My question: is there a generalized result for pseudo metric spaces embedding isometrically into semi-...
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 ...
13 votes
2 answers
767 views

Smooth Urysohn's lemma on Fréchet spaces

Let $V$ be a Fréchet topological vector space. Let $K_0$ and $K_1$ be two closed subsets which are disjoint. I wish to show the existence of a Fréchet-smooth function $f:V\to [0,1]$ whose restriction ...
1 vote
0 answers
178 views

Trans-universality for finite-dimensional Banach space

In addition to a specific problem Trans-universality for finitely generated groups, I posted also its general form. It should not hurt to provide another special case: QUESTION: does there exist a ...
1 vote
1 answer
284 views

Bilipschitz embedding of the unit ball of $c_0$ into $\ell_1$

This is a follow-up to this question of mine. It is well-known that the Banach space $\ell_1$ does not contain any isomorphic copies of $c_0$. One can even go further and show that $\ell_1$ does not ...
0 votes
1 answer
232 views

A generalization about the density of $\mathcal C_c(X, E)$ in $\mathcal L_p (X, \mu, E)$ when $E$ is a Banach space

Let $X$ be a metric space, $\mu$ a $\sigma$-finite non-negative Borel measure on $X$, and $(E, |\cdot|)$ a Banach space. Let $\mathcal L_p := \mathcal L_p (X, \mu, E)$ and $\|\cdot\|_{\mathcal L_p}$ ...
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 ...
1 vote
1 answer
61 views

Are there some conditions on a metric space $X$ such that these two types of weak converge of finite signed Borel measures on $X$ are related?

Let $X$ be a metric space, $\mathcal M(X)$ the space of all finite signed Borel measures on $X$, and $\mathcal C_b(X)$ be the space of real-valued bounded continuous functions on $X$. Then $\mathcal ...
0 votes
1 answer
55 views

Get an estimate on $L^{2}(0,1)$ [closed]

Consider $f \in L^{2}(0,1)$ and $g \in L^{\infty}(0,1)$ such that $ \text{lim} ~g(x) = 0 \ \ \text{when} \ \ x \to 0^{+};$ $g(x) > 0 \ \forall x \in (0,1)$; $\text{lim}~\dfrac{g(x)}{x^{\alpha}} =...
3 votes
0 answers
115 views

Isometric embeddings of $c_0$ into metric spaces

Are there any nice and useful criteria or theorems which assert when a given metric space $M$ contains an isometric (not necessarily linear) copy of the Banach space $c_0$ or its unit ball $B_{c_0}$? (...
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). ...
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_{\...
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-...
3 votes
0 answers
165 views

Which metric spaces embed isometrically in $\ell_p$?

It is known that each metric space $X$ embeds isometrically in the Banach space $\ell_\infty(X)$ of bounded (not necessarily continuous) functions $X \to \mathbb R$. Since $\ell_\infty(X)$ does not ...
1 vote
1 answer
114 views

Example of a nonconvex Chebyshev set in a metric space with continuous projection?

Question: Is there an example of a nonconvex Chebyshev set $S$ in a metric space $(X,d)$ whose projection map is continuous? For convexity to be well-defined, we need to assume that $X$ is a vector ...
0 votes
1 answer
407 views

Criteria for $\epsilon$-Density

Let $Y$ be a compact, separable metric space and $X=C(Y)$ Banach space. There are many criteria for a linear subspace $Z\subseteq X$ to be dense; notably the Stone-Weierstraß theorem. Are there ...
1 vote
1 answer
896 views

Known Lipschitz-free spaces

The Lipschitz-Free space (also known as Arens-Eells spaces) $\mathcal{F}(X,d)$ over a pointed metric space $(X,d)$ is a well-studied object. In many instances, we have "concrete" representations of ...
2 votes
2 answers
264 views

Existence of a Hölder-free space

The Lipschitz-free or Arens-Eells space over a pointed separable metric space $(X,0,d)$ is a well-studied object. My question is, is an analogos Hölder-free space; for a fixed Hölder constant $\alpha&...
1 vote
1 answer
107 views

Continuous Left-inverse of Dirac Lipschitz-Free Space

Let $X$ be a separable pointed metric space and let $AE(X)$ denote the corresponding Lipschitz-Free (or Arens-Eells space) over $X$. The point-evaluation map $\delta:X\mapsto AE(X)$ is injective ...
1 vote
1 answer
791 views

$\ell_1$ and $\ell_\infty$ as complementary subspaces of Banach space

Let $X$ be a Banach space, and let $X'\subset X$ - its subspace. Then the following propositions are true: $X'$ is closed, $X/X' \cong \ell_1 \Rightarrow X'$ is complementary; $X' \cong \ell_\infty ...
3 votes
1 answer
404 views

Ultraproduct of metric spaces

Let $I$ be a set and $\mathcal{U}$ be an ultrafilter on $I$. Suppose that $(X_{i}, d_{i})_{i\in I}$ is a family of pointed metric spaces with a distinguished point $e_{i}$ for each $i\in I$. We set $$...
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 ...
7 votes
0 answers
172 views

Choice and the Baire property in non-separable complete metric spaces

It's known to be consistent with ZF+DC that every subset of $\mathbb{R}$ has the Baire property (BP). (E.g. Shelah's model). If so, then every subset of every complete separable metric space has ...
4 votes
2 answers
309 views

Finitely isometrically persistent metric spaces

The goal of this question is to develop further the discussion initiated in Under which conditions is it possible to find points with same distances under bi-Lipschitz map. The mentioned question was ...
3 votes
1 answer
807 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 ...
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-...