## Characterization of separability

Prove that a metric space is separable if and only if there does not exist an uncountable set such that for every .

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 a segment () of and a closed linear subspace of such that each function in is of class on . For such that we denote where .

(a) Prove that .

(b) Prove that for every we have

(c) Deduce that there exists which verifies for each and

(d) Let be a finite sequence of points in such that (for ) and . Prove that

2. Let be a closed linear subspace of such that every function in is of class on . Prove that has finite dimension.

## l-infinity is not separable

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

## Separable space 2

Denote , the set of complex sequences which converge to . Furthermore, consider the sequences . Prove that the closed linear span of is in fact .

## non-Separable space Example 1

Prove that the space is not separable.

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## Separable spaces

A Banach space is called separable if it contains a countable dense subset. Here are some interesting facts about separable spaces.

Prove that if is a subset of a separable metric space then is also separable.

If the dual of a normed vector space is separable, then the space itself is separable.