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So, I know P. Enflo showed that there is a separable Banach Space that doesn't satisfy the approximation property. My professor mentioned during class that in fact generic separable Banach Spaces don't have a Schauder basis where generic is in terms of Baire Category with some topology. I was hoping someone could outline how we put a topology on the space of all Banach Spaces.

From what I know about these types of proofs, most likely we show that our set is dense $G_{\delta}$ and so comeager. The $G_\delta$ portion I assume is the standard argument, but I'm curious as to how we show density.

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    $\begingroup$ I'd be interested to know the answer. There is a way of putting a separable metrizable (Polish) topology on SB (the collection of all separable Banach spaces). With this set up it can be shown that the collection of a spaces with Schauder bases is analytic but it it not known if it is non-Borel. If one could show this collection is non-Borel that would imply that there is a separable space without a basis without needing Enflo's example, however, I don't think this has been proven. If I remember correctly (talking to Pandelis Dodos 3 years ago) Johnson asked him this question. $\endgroup$ Oct 13, 2015 at 16:25
  • $\begingroup$ @Kevin Beanland: "There is a way of putting a separable metrizable (Polish) topology on SB (the collection of all separable Banach spaces). With this set up ..." Do you have a reference for this? Isn't this precisely (at least partly) what the OP is asking: "I was hoping someone could outline how we put a topology on the space of all Banach Spaces." $\endgroup$
    – TaQ
    Oct 14, 2015 at 16:33
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    $\begingroup$ A good reference is Pandelis Dodos' Book "Banach Spaces and Descriptive Set Theory: Selected Topics" found here: users.uoa.gr/~pdodos/research.html $\endgroup$ Oct 14, 2015 at 16:59
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    $\begingroup$ @KevinBeanland That book seems right along the lines of what I was looking for. I guess I'll start working through it since Banach Spaces and DST are both super interesting. $\endgroup$ Oct 15, 2015 at 2:09
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    $\begingroup$ You are at the right place for your interests, Konrad. $\endgroup$ Oct 15, 2015 at 20:20

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