Despite over one-hundred applicants our math department never managed to fill the two open positions we had (we made three offers). Since they really don’t want to hire anyone who is not likely to get tenure we never went to a secondary pool. As such, they sucked a couple of us from the physics department into teaching a math course or two (with paid overload, of course). So I have the interesting task of teaching both Linear Analysis (which is just plain easy) as well as Real Analysis this semester. Since the math chair is likely to be on sabbatical in the spring, I may even end up as acting chair of that department for the spring semester (lucky me??).
In any case, both courses are going well so far and I will probably teach the second semester of Real Analysis in the spring (technically the course title is Advanced Calculus I & II). I’m using the books Understanding Analysis by Stephen Abbott and The Way of Analysis by Robert Strichartz. The former is an excellent and amazingly readable text, but only covers one semester. The latter is readable but at a different level, and can be obtuse in spots (and, as a cheaply made paperback, is obscenely overpriced). But it is the best two-semester book I could find. So I’m using both. I have also recommended that students have, as a reference, Jan Gullberg’s Math: From the Birth of Numbers. I love that book. Great reference. While I was in Montana, Patrick Hayden recommended as an additional reference, J. Michael Steele’s The Cauchy-Schwarz Master Class. I haven’t picked it up yet, but based on the excerpt on Amazon, I like his thinking. The opening line is:
Cauchy’s inequality for real numbers tells us that and there is no doubt that this is one of the most widely used and most important inequalities in all of mathematics.
From the Cauchy-Schwarz inequality one can derive the operator form of the uncertainty principle and one may also derive the ubiquitous triangle inequality. From the latter, one may then derive a form of Bell’s inequalities.
For Linear Analysis, I’m using a free text written by Jim Hefferon at Saint Michael’s College in Vermont (one of our chief competitors and the school that is probably most like us). My biggest complaint is the numbering for the section headings and his penchant for somewhat odd notation in spots. Otherwise, I think it is a great book. In fact I’d say it’s better than Anton’s famous book on the subject which is what I used when I was in college.