Prove the following remarkable properties of and

- Given any normed space and a subspace , any continuous linear map may be extended to all of preserving its norm.
- Every separable Banach space is a quotient of

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# Universal and injective Banach spaces

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Prove the following remarkable properties of and

- Given any normed space and a subspace , any continuous linear map may be extended to all of preserving its norm.
- Every separable Banach space is a quotient of

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The first claim is an application of the Hahn Banach theorem. Indeed, consider the dual functoinals of and note that for any , we have , take and extend them with Hahn Banach, say to . Then defined by solves the problem.

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

1.

2.

3.

the last equality being true since 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 is a quotient map. Given , we can choose such that . Set . We show that for some in the open unit ball of of radius . Take . Since is in the ball, there is some such that . Now is in which is the closure of for ; and we can find such that . Continue inductively, and obtain that . Then is the image of which has norm .

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