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Results tagged with metric-spaces
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user 6101
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) $d(x,y)+d(y,z) \geq d(x,z)$.
11
votes
Smooth Urysohn's lemma on Fréchet spaces
A bump function on a Banach space $X$ is a non-zero function with bounded support. Existence of a bump function of a given smooth regularity has, of course, strong immediate consequences --by translat …
- 60.6k
1
vote
Accepted
What is the quotient (pseudo)metric $d_\sim$ and how do I identify the infimum of possible s...
One gets the trivial semi-distance. Let $d$ denote the semi-distance on $Z$ that defines $d_\sim$ on $Z/\sim$, that is $d_\sim([z],[z']):=d(z,z')$ for all $z$ and $z'$ in $Z$. Thus $d(z,z')\le |z-z'|$ …
- 60.6k
3
votes
Accepted
$\ell_1$ and $\ell_\infty$ as complementary subspaces of Banach space
For (1), use (0): for any Banach space $X$, $L(\ell_1,X)$ is isometrically isomorphic to the space $\ell_\infty(X)$ of bounded sequences in $X$. You may easily define a concrete isometry and its inver …
- 60.6k
15
votes
Converse to Banach's fixed point theorem?
If you want a sort of positive result, this comes to my mind, for what is worth:
If $X$ is a metric space such that any
contraction map $T:Y \to Y$ on any nonempty
closed subset of X has a f …
- 103
11
votes
Accepted
Metrization of spaces of functions
As to the compact-open topology of $C(X,Y)$, it is metrizable if and only if $Y$ is metrizable, and $X$ is hemicompact.
- 60.6k
5
votes
Accepted
Two metrics and a sequence converging to two points.
It is a "bad question", in that the assumption that neither metric topology is contained in the other does not imply that there is a sequence converging in both topologies, but to different points. Ex …
- 60.6k
7
votes
How to show the cardinality of nonisometric compact metric spaces is the continuum
Since a compact metric space is in particular separable, its type of isometry is determined by a dense countable subspace. There are continuum many distances on , say $\mathbb{N}$.
- 60.6k
5
votes
Measure of the support of a Borel probability on a metric space
Consider an uncountable discrete metric space $X $ (i.e., metrized by the Kronecker delta). Define a measure on $X$ putting for any $A\subset X,\\ $ $\mu(A)=1$ or $\mu(A)=0$ according whether $A$ belo …
- 60.6k