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

 

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Universal and injective Banach spaces

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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