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.