2) In the proof of Corollary, it is “Theorem 2” instead of “Theorem 1”.

3) In Corollary 1, a “Tx” is not in mathematical mode.

*[Corrected, thanks. I do not see a quick way to recover the implication of 1 from 3, so it is probably best to just delete the remark. -T.]*

*[Corrected, thanks – T.]*

What is the Hahn-Banach Theorem used for in Exercise 6?

*[It is not used, and the reference to it has been removed – T.] *

*[ here should have been – T.]*

*[Corrected, thanks – T.]*

*[Corrected, thanks – T.]*

The hypothesis 1 is nothing more than the assertion that the cover , and *thus by the Baire category theorem must be dense in a ball*.

Do you mean that one of the (instead “all of them”) must be dense in some ball? Or do you mean (trivially) that the union of them is dense in a ball?

*[Corrected, thanks – T.]*