Question about quotient of separable spaces

Asked Nov 02 '20 at 01:47

Active Nov 02 '20 at 01:57

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Exercise: Let $E$ a Banach space separable and $F$ a closed subspace of $E$. Prove that $E/F$ is separable.

My idea: Since $E$ is separable then exist $D_1\subset E$ numerable and dense and Since F is closed then $F=\bar{F}$. Moreover $F$ is separable, beacuse every subset of separable space is separable.

We know that \begin{equation*} E/F=\{[x]: x\in F\} \end{equation*}

Maybe, We can find a numerable and dense subset consisting of equivalences clases. But, I don't know how to relate the quotient $E/F$ in the proof.

edited Nov 02 '20 at 01:57

J. W. Tanner

60,406

asked Nov 02 '20 at 01:47

MCM

1

If $(x_n){n \in \Bbb{N}}$ is dense in $E$, then $([x_n]){n \in \Bbb{N}}$ is dense in $E/F$. See if you can prove it! – user837206 Nov 02 '20 at 03:01
Hint: $||[x_n]|| \leq ||x_n||$ – Evangelopoulos Foivos Nov 02 '20 at 04:50
I would like to mention that subsets of separable metric spaces need not be separable (this fact is usually expressed saying that separability is not hereditary). However, this is true for closed subspaces in separable Banach spaces. – Davide Ravasini Nov 02 '20 at 09:31
@DavideRavasini subsets of separable metric spaces are separable. – Gabriel Romon Nov 29 '21 at 15:50

Question about quotient of separable spaces

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