Every Banach space is quotient of $\ell_1(I)$

Question

I'm looking for a book containing the proof that for every Banach space E there is an index I so that E is a quotient space of $\ell_1(I)$. If I can't find the book on google books, it would be great if you could give me the page number, because my paper is due tomorrow and I just need it for a reference. Thank you!

Answer 1

Lemma 1.4.a in:

J. M. F. Castillo and M. Gonzalez, Three-space problems in Banach space theory, Springer Lecture Notes in Math. 1667, 1997.

Proof. Let $E$ be a Banach space and let $\{x_i\colon i\in I\}$ be a dense subset of the unit ball of $E$. Define a map $Q\colon \ell_1(I)\to E$ by

$$Q ( (a_i)_{i\in I} ) = \sum_{i\in I} a_i x_i\quad (a_i)_{i\in I} \in \ell_1(I).$$

Then $Q$ is the desired surjection. $\square$

Answer 2

You need the Banach space to be separable, I think.

You can find it as Theorem 4.6 in these notes by Piotr Hajlasz.