# Universal and injective Banach spaces

Prove the following remarkable properties of $\ell_1$ and $\ell_\infty$

1. Given any normed space $X$ and a subspace $Y$, any continuous linear map $T:Y\to\ell_\infty$ may be extended to all of $X$ preserving its norm.
2. Every separable Banach space is a quotient of $\ell_1$

## One thought on “Universal and injective Banach spaces”

1. The first claim is an application of the Hahn Banach theorem. Indeed, consider the dual functoinals $\hat e_n$ of $\ell_\infty$ and note that for any $y\in Y$, we have $Ty = (\hat e_n Ty) = (\hat y_n(y))$, take $\hat y_n\in Y'$ and extend them with Hahn Banach, say to $\hat x_n \in X'$. Then $S:X\to \ell_\infty$ defined by $Sx = (\hat x_n(x))$ solves the problem.

For the second, take a dense countable subset $(x_n)$ of the unit ball in $X$ a separable space, and define an obvious map $Q:\ell_1\to X$ that sends $e_n\to x_n$. This gives a map $Q$ with norm at most $1$ such that

1. $Q(B_{\ell_1}^\circ) \subseteq B_X^\circ$
2. $Q(B_{\ell_1}) \subseteq B_X$
3. $\overline{Q(B_{\ell_1})}=B_X$

the last equality being true since $(x_n)$ was chosen to be dense.

Now we mimic the proof of the open mapping theorem to show that the open unit ball is sent to the open unit ball, and conclude $Q$ is a quotient map. Given $x\in B_X^\circ$, we can choose $\varepsilon$ such that $|x| <1-\varepsilon <1$. Set $y =(1-\varepsilon)^{-1} x$. We show that $Qz=y$ for some $z$ in the open unit ball of $\ell_1$ of radius $(1-\varepsilon)^{-1}$. Take $\delta <\varepsilon$. Since $y$ is in the ball, there is some $n_1$ such that $|y-x_{n_1}|<\delta$. Now $y-x_{n_1}$ is in $\delta B_X$ which is the closure of $\delta x_n$ for $n\neq n_1$; and we can find $n_2$ such that $|y-x_{n_1}-\delta x_{n_2}| <\delta^2$. Continue inductively, and obtain that $y = \sum_{j\geqslant 1} \delta^{j-1} x_{n_j}$. Then $y$ is the image of $\sum_{j\geqslant 1} \delta^{j-1} e_{n_j}$ which has norm $(1-\delta)^{-1} <(1-\varepsilon)^{-1} < 1$.

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