Assorted notes on functional analysis

An introduction for functional analysis

Notes created for a course on Functional analysis. I wrote these up as separate notes for the spring term of 2004; for the spring term of 2005, I have collected the notes into one document. For the spring term of 2006 I corrected a number of misprints, improved the exposition a few places, and added some more material in the section on spectral theory.

Although these notes are no longer used locally, I do make some alterations now and then when I feel like it, just in case someone out in the world is interested. In the latest iteration, I added proper attribution to Goldstine's theorem, improved the proof of Lemma 68 (previously Lemma 67), and I added Kakutani's theorem, since I could get it almost for free.

This version is dated 2022-02-23.

The main book for the course is Kreyszig's Intoductory Functional Analysis With Applications. These notes serve as a supplement.

Contents:

  1. Transfinite induction
  2. Some Banach space results (alternate proof of Banach–Steinhaus)
  3. Sequence spaces and Lp spaces
  4. A tiny bit of topology
  5. Topological vector spaces
  6. Spectral theory
  7. Compact operators

To view or download

Best for on-screen viewing: A5 size (PDF).

Best for printing: A4 size (PDF).
(If you need to print on US letter paper, be sure to ask your printer driver to reduce the size to fit the paper.)

Credits

Thanks to Haixing Hu for pointing out some mistakes in an earlier version of the chapter on transfinite induction. Also thanks to Xavier Raynaud for finding several points in the same chapter where my exposition was much too brief. The improved proof of Lemma 68 (February 2022) is due to an observation made by Anh Dũng Lê.

Please respect the copyright

The notes may be freely copied for personal or classroom use; contact me if you wish to copy them for any other reason.