The present book has been written so as to necessitate as little consultation by the reader as reasonably possible of other published material. I have hoped to thereby make it accessible to people far from large research centres or any ‘good library”, and to those who have only their summer vacations to work on mathematics. It is for the same reason that references, where unavoidable, have been made to books rather than periodicals whenever that could be done.

In general, I consider the developments leading up to the various results in the book to be more important than the latter taken by themselves; that is why those developments are set out in more detail than is now customary. My aim has been to enable one to follow them by mostly just reading the text, without having to work on the side to fill in gaps. The reader's active participation is nevertheless solicited, and problems have been given. These are usually accompanied by hints (sometimes copious), so that one may be encouraged to work them out fully rather than feeling stymied by them. It is assumed that the reader's background includes, beyond ordinary undergraduate mathematics, the material which, in North America, is called graduate real and complex variable theory (with a bit of functional analysis). Practically everything needed of this is contained in Rudin's well-known manual.