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Results tagged with metric-spaces
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user 121665
A metric space is a pair $(X,d)$, where $X$ is a set and $d:X \times X \to \mathbb{R}$ satisfies the following conditions for all $x,y,z \in X$. (Symmetry) $d(x,y)=d(y,x)$. (Identity of Indiscernibles) $d(x,y)=0$ if and only if $x=y$. (Triangle Inequality) $d(x,y)+d(y,z) \geq d(x,z)$.
17
votes
What is the structure preserved by strong equivalence of metrics?
Metrics are strongly equivalent if the identity mapping $Id:(X,d_1)\to (X,d_2)$ is bi-Lipschitz. They preserve the class of Lipschitz mappings.
Roughly speaking classical topology deals with notions …
- 28k
8
votes
Uniform density of Lipschitz maps is space of continuous function — for general metric spaces
While in general one cannot expect density of Lipschitz mappings (as pointed out by Nik Weaver), the following classical result is in the positive direction. You can find a proof in Heinonen's book Le …
- 28k
5
votes
Accepted
Domains in $\mathbb{R}^n$ for which Hajlasz-Sobolev spaces and Sobolev Spaces are the same
Your argument is not correct. If a property $P$ fails for $Y$ and $X\subset Y$, it does not follow that it fails for $X$. For example $X=\{0\}\subset\mathbb{R}=Y$ but there are many properties true fo …
- 28k
5
votes
Accepted
Reference: Hajlasz-Sobolev Spaces with Values in a Metric Space
The question is not stated in a very clear manner, but nevertheless, the answer is: no.
Separability. The space $M^{1,p}(X,d,\mu)$ is not separable even if $X$ is the standard ternary Cantor set, $d$ …
- 28k
4
votes
Accepted
Continuous inclusion of metric spaces of smaller capacity
The answer is no. You can have $X$ path connected and $Y$ totally disconnected satisfying your condition. Then any continuous map from $X$ to $Y$ is necessarily constant so there are no injective maps …
- 28k
3
votes
Accepted
Does there always exists a path $g:[0,1] \rightarrow X$ from $f(0)$ to $f(1)$ that has the s...
This is a standard result that every rectifiable curve in a metric space admits an arc length parametrization. The proof can be found in many sources. For examples Theorem 3.2 in
Hajłasz - Sobolev spa …
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3
votes
Accepted
Monotonicity of doubling dimension
This is Lemma 9.6(i) in J. C. Robinson, Dimensions, embeddings, and attractors. Cambridge Tracts in Mathematics, 186. Cambridge University Press, Cambridge, 2011.
In the proof the author says "it is …
- 28k
1
vote
Accepted
Is the function $g$ always injective where $g$ is obtained by lipschitz re-parametrization
It is injective because if $g(s)=g(t)$, $s<t$, then you can remove the interval $[s,t]$ from the domain of definition of $g$ and make the curve shorter. Then you rescale the domain to be $[0,1]$. Resc …
- 28k