### Archive

Posts Tagged ‘separable’

## Characterization of separability

Prove that a metric space ${X}$ is separable if and only if there does not exist an uncountable set ${S \subset X}$ such that ${d(x,y)\geq c>0}$ for every ${x,y \in S,\ x \neq y}$.

This is a useful property which makes trivial the proof of the fact that any subset of a separable space is separable.

## Agregation 2013 – Analysis – Part 4

Part IV: To be or not to be with separable dual

1. Consider ${S=[a,b]}$ a segment (${a) of ${[0,1]}$ and ${V}$ a closed linear subspace of ${C([0,1])}$ such that each function in ${V}$ is of class ${C^1}$ on ${S}$. For ${(x,y) \in S^2}$ such that ${x \neq y}$ we denote ${\displaystyle \xi_{x,y}(f)=\frac{f(x)-f(y)}{x-y}}$ where ${f \in V}$.

(a) Prove that ${\xi_{x,y} \in V^*}$.

(b) Prove that for every ${f \in V}$ we have

$\displaystyle \sup_{\substack{x,y \in S\\ x\neq y}} |\xi_{x,y}(f)|<\infty.$

(c) Deduce that there exists ${\mathcal{N}(S)>0}$ which verifies for each ${f \in V}$ and ${(x,y) \in S^2}$

$\displaystyle |f(x)-f(y)| \leq \mathcal{N}(S) |x-y|\|f\|_\infty.$

(d) Let ${(t_l)_{0\leq l \leq L}}$ be a finite sequence of points in ${S}$ such that ${0 (for ${0\leq l ) and ${t_0=a,t_L=b}$. Prove that

$\displaystyle \forall f \in V,\ \sup_{t \in S}|f(t)| \leq \sup_{0\leq l\leq L}|f(t_l)|+\frac{1}{2}\|f\|_\infty$

2. Let ${F_0}$ be a closed linear subspace of ${C([0,1])}$ such that every function in ${F_0}$ is of class ${C^1}$ on ${[0,1]}$. Prove that ${F_0}$ has finite dimension.

## l-infinity is not separable

Denote by $\ell^\infty$ the space of all bounded complex(real) sequences. Prove that this space is not separable.

Categories: Functional Analysis Tags:

## Separable space 2

Denote $c_0=\{(\alpha_n) \subset \Bbb{C} : \alpha_n \to 0\}$, the set of complex sequences which converge to $0$. Furthermore, consider the sequences $x_n=((n+1)^{-j})_j$. Prove that the closed linear span of $\{x_n : n=1,2,..\}$ is in fact $c_0$.

## non-Separable space Example 1

Prove that the space $C=\{f: [1,\infty) \to \Bbb{C},\ f \text{ continuous and bounded }\}$ is not separable.
A Banach space $X$ is called separable if it contains a countable dense subset. Here are some interesting facts about separable spaces.
Prove that if $Y$ is a subset of a separable metric space $(X,d)$ then $Y$ is also separable.
If the dual of a normed vector space $X$ is separable, then the space itself $X$ is separable.