ChapterZero

I’ve been doing a lot less reading than I intended to, but lately I’ve been focusing what willpower I have at looking at Ward Cheney’s book “Analysis for Applied Mathematics”. I’ve been skipping about, skimming whatever catches my attention; fortunately, he writes in neat packets which can be digested in such a manner. The first thing I read (well, skimmed) was his discussion of a homotopic method used to solve linear programming problems with equality constraints– this all boils down to some clever matrix manipulations that really don’t have much to do with homotopy, but it’s still nice to see that such a practical algorithm can be derived from such a seemingly abstract concept.

Today, I read and reconstructed his proof of the Hahn-Banach theorem with great pleasure: his is a much clearer exposition than the one provided by Folland, which introduced me to the theorem. Whereas, if I recall correctly, Folland starts off by proving a mysterious inequality and then shows that it implies the extension theorem, Cheney does an excellent job of motivating the inequality, and the proof naturally falls out along the way. Also, Folland uses sups and such as an integral part of his argument, while Cheney avoids limit arguments except for at one crucial point, where the limiting argument is so obvious that it wasn’t until I was reconstructing his argument that I noticed the omission.