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# Erdman John M. Companion to Real Analysis

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Unpublished, 2012. — 265 p.
Paul Halmos famously remarked in his beautiful Hilbert Space Problem Book [22] that \The only way to learn mathematics is to do mathematics." Halmos is certainly not alone in this belief.
The current set of notes is an activity-oriented companion to the study of real analysis. It is intended as a pedagogical companion for the beginner, an introduction to some of the main ideas in real analysis, a compendium of problems I think are useful in learning the subject, and an annotated reading/reference list.
The great majority of the results in beginning real analysis are straightforward and can be verified by the thoughtful student. Indeed, that is the main point of these notes|to convince the beginner that the subject is accessible. In the material that follows there are numerous indicators that suggest activity on the part of the reader: words such as :proposition", "example", "exercise", and "corollary", if not followed by a proof or a reference to a proof, are invitations to verify the assertions made. Of course, the proofs of some theorems, the Hahn-Banach theorem for example, are, in my opinion, too hard for the average student to (re)invent. In such cases when I have no improvements to offer to the standard proofs, instead of repeating them I give references.
These notes were written for a year long course in real analysis for seniors and rst year graduate students at Portland State University. During the year students are asked to choose, in addition to these notes, other sources of information for the course, either printed texts or online documents, which suit their individual learning styles. As a result the material that is intended to be covered during the Fall quarter, the rst 10{15 chapters, is relatively complete. After that, when students have found other sources they like, the notes become sketchier.
I have made some choices that may seem unusual. I have emphasized the role played by order relations in function spaces by trying to make explicit the vector lattice structure of many important examples. This material usually seems to have some \poor relation" status in most real analysis texts. To highlight fundamental concepts and avoid, on a first acquaintance, distracting technical details I have treated compact spaces more thoroughly than locally compact ones and real measures more thoroughly than extended real valued ones.
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