These are the lecture notes for a course I gave on the elementary theory of Hp spaces at the Stockholm Institute of Technology (tekniska högskolan) during the academic year 1977–78. The course concentrated almost exclusively on concrete aspects of the theory in its simplest cases; little time was spent on the more abstract general approach followed, for instance, in Gamelin's book. The idea was to give students knowing basic real and complex variable theory and a little functional analysis enough background to read current research papers about Hp spaces or on other work making use of their theory. For this reason, more attention was given to techniques and to what I believed were the ideas behind them than to the accumulation of a great number of results.

The lectures, about Hp spaces for the unit circle and the upper half plane, went far enough to include interpolation theory and BMO, but not as far as the corona theorem. That omission has, however, been put to rights in an appendix, thanks to T. Wolff's recent work. His proof of the corona theorem given there is a beautiful application of some of the methods developed for the study of BMO.

For Carleson's original proof of the corona theorem the reader may consult Duren's book. I have not included the more recent applications of the geometric construction Carleson devised for that proof, such as Ziskind's.