How to prove the following mind-blowing fact?
Let $X$ be a separable Banach space and let $\ell_1$ be the space of all absolutely summable scalar sequences. Then there exists such closed subspace $A\subset \ell_1$ that factor space $\ell_1/A$ and $X$ are isomorphic as normed spaces.
Edit:
So what, this is like a classification up to isomorphism of all separable Banach spaces? Each separable Banach space corresponds to some closed subspace of $\ell_1$?
Best Answer
Let $X$ be a Banach space and let $\{x_d\colon d\in D\}$ be a dense subset of the unit ball of $X$. Consider the space $\ell_1(D)$ of all absolutely summable sequences on $D$. We define a linear map $T\colon \ell_1(D) \to X$ by
$$T\Big((\lambda_d)_{d\in D}\Big) = \sum_{d\in D}\lambda_d x_d\qquad ((\lambda_d)_{d\in D} \in \ell_1(D)).$$ This is a well-defined linear map as the right-hand side converges absolutely for every $(\lambda_d)_{d\in D}\in \ell_1(D)$, hence it defines an element of $X$. For each $(\lambda_d)_{d\in D} \in \ell_1(D)$ we have $$\|T\big((\lambda_d)_{d\in D}\big)\|\leqslant \sum_{d\in D}\|\lambda_d x_d\|\leqslant \sum_{d\in D}|\lambda_d|=\|(\lambda_d)_{d\in D} \|.$$
Consequently, $T$ is a bounded (actually norm-one) linear operator. Since $\{x_d\colon d\in D\}$ is dense in the unit sphere of $X$, $T$ is surjective. By the first isomorphism theorem, $$X\cong \ell_1(D) / \ker T.$$
Note that separability of $X$ means that we may take $D=\mathbb{N}$.
Here's a reference to the literature.