for Math 648

These notes are intended to merely augment and not completely reproduce the recorded lectures. I think it's can be a little tough to see what he's writing, so I think this will be helpful. Professor Benedetto gives significant informative exposition during his lectures (which I appreciate). I have to apologize, but I have largely left this out. This is not out of disrespect, but it's hard to do it much justice without pictures (which I'm not drawing here), and my notes on this are poor (I'm just listening). I think this is much better understood through the videos, anyways. A note on errors: I'd like these to be professional quality, but I'm producing these notes for my own purposes. I have handwriting like a three year old, but I like to keep my lecture notes. I have absolutely no doubt that this document is riddled with typesetting errors. I welcome your help in finding them. Please email [email protected], and I'll happily fix them. This will help me, and hopefully you too. I have tried to stick with his notation. I apologize for the places that I've failed, this is not out of disrespect, just simply my bad habits. I have taken liberty on occasion. In particular, theorems quoted for the result with no proof given I have called propositions, and I have pulled some things I call lemmas out of the proofs to streamline. Any other places that I have taken liberty I explicitly mention. I have cited several things in " Real and Complex Analysis " by Walter Rudin [7], which is the only real analysis book that I have. These things can no doubt be found in any decent analysis book, so consult what you've got.