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Results tagged with metric-spaces
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user 2554
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) $d(x,y)+d(y,z) \geq d(x,z)$.
14
votes
Accepted
covering a separable metric space by small balls
The answer is no for the Banach space $c_0$. Suppose $B(x_i,r_i)$ is a sequence of balls with $r_i\to 0$ and WLOG $x_i$ is supported in $[1,N_i]$ with
$N_1<N_2<...$. Consider a point $x$ in $c_0$ w …
- 31.5k
5
votes
Accepted
Equivalent metrics on Fréchet spaces and Lipschitz maps
Use $\sum 2^{-n}(\|x-y\|_n \wedge 1)$ for the distance on $Y$ and $\sum 2^{-n}(\|x-y\|_n \wedge 2)$ for the distance on $X$.
- 31.5k
5
votes
Accepted
Is the Hausdorff metric on sub-$\sigma$-fields separable?
Take a sequence $A_n$ of independent sets of measure $1/2$. Given two different subsets $B$ and $C$ of natural numbers, suppose WLOG that there is an $n$ in $B\sim C$. Now $\mu(A_n\Delta A) = 1/2$ f …
- 31.5k
10
votes
Accepted
In infinite dimensions, is it possible that convergence of distances to a sequence always im...
That every Banach space is contained in a $P$-complete Banach space follows immediately from the following
Theorem.
Let $X$ be a Banach space. Then there exists a Banach space $Y$ containing $X$ in wh …
- 31.5k