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Posts Tagged ‘separable’

Characterization of separability

September 15, 2013 Leave a comment

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.

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Agregation 2013 – Analysis – Part 4

June 24, 2013 Leave a comment

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

1. Consider {S=[a,b]} a segment ({a<b}) 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<t_{i+1}-t_i\leq \displaystyle\frac{1}{\mathcal{N}(S)}} (for {0\leq l <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.

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l-infinity is not separable

January 8, 2011 2 comments

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

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Separable space 2

January 4, 2011 2 comments

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.

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non-Separable space Example 1

January 4, 2011 Leave a comment

Prove that the space C=\{f: [1,\infty) \to \Bbb{C},\ f \text{ continuous and bounded }\} is not separable.
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Separable spaces

January 4, 2011 8 comments

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.

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