I'm really confused with this question...

Let $X$ be a separable Banach space. Prove that $X$ is quotient space $\ell_1$. I have some denotations.

$\{ \xi_i \}$ - convergent dense sequence in $K_1(0)$ = $\{ x \in X \mid \|x\| \leq 1 \}$ - closed balls with center $0$.

$ \alpha : \{ x_i \} \mapsto \sum_i (x_i \xi_i),\{ x_i \} \in \ell_1 $

How can these denotations help me to prove statement? Thanks in advance.

  • 3
    $\begingroup$ You're giving no hypotheses on $X$. Perhaps you want to show every separable Banach space is a quotient of $\ell_1$. For a very nice proof, check Carother's book "A short course on Banach Space theory", Theorem 6.1. Essentially, one picks a dense sequence $(x_n)$ in $B_X$ and defines a map $\ell_1 \to X$ so that $e_i\mapsto x_i$ that extends linearly to a continuous map $Q$ of norm at most one. Then one works in a similar fashion than that of the open mapping theorem to show this map is open and surjective: it suffices you show $\overline{Q(B_{\ell_1})}=B_X$. $\endgroup$ – Pedro Tamaroff Oct 28 '16 at 5:08