All Questions
Tagged with fa.functional-analysis metric-spaces
78 questions
3
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0
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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 ...
- 63.9k
5
votes
2
answers
642
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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 $\mathcal{F}$ ...
- 3,312
0
votes
1
answer
407
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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 ...
- 2,263
1
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0
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84
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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 ...
- 63.9k
1
vote
1
answer
896
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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 ...
- 2,472
3
votes
1
answer
148
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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
$...
- 12.6k
0
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0
answers
45
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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
1
vote
1
answer
107
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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 ...
- 3,486
6
votes
1
answer
348
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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}(...
- 28k
4
votes
0
answers
99
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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\...
- 3,486
2
votes
1
answer
1k
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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
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
1
vote
1
answer
790
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$\ell_1$ 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 ...
- 60.5k
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 ...
- 538
1
vote
0
answers
105
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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
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
1
vote
1
answer
162
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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}\...
- 623
2
votes
1
answer
87
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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\ $?
- 66.3k
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\ \...
CommunityBot
- 1
3
votes
1
answer
806
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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 ...
- 4,878
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 $...
- 6,210
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-...
- 6,206
2
votes
0
answers
126
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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
5
votes
1
answer
148
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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
1
vote
1
answer
217
views
Open set of geodesics implies the set of starting points is open
Let $X$ be a complete and separable metric space, let $G(X) \subset C([0,1],X)$ be the space of continuous curves from $[0,1]$ to $X$ with constant speed, i.e.
$$ d(f(t),f(s)) = |t-s| d(f(0), f(1)). $$...
- 41
13
votes
1
answer
4k
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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 ...
CommunityBot
- 1
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 ...
- 31.5k
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 ...
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