As Rigorous As It Gets

As a math undergrad, there have been quite a few instances, where I had to stare at a single line in an analysis proof for hours, because the author had skipped too many “obvious” steps as an exercise for the reader. While I am glad the author did it (for it makes the reader a better mathematician), I wish to write about those “gaps” in common analysis proofs (just because I can). I also wish to provide some extensions/ generalizations of certain proofs and certain applications (in the form of theorems) in other fields that follow from tools in real analysis.

Here, I have written extensions/ explanations of proofs of theorems from mostly three books:

  • Gerard B. Folland’s “Real Analysis: Modern Techniques and Applications” (Graduate Real Analysis)

  • Walter Rudin’s “Functional Analysis” (Graduate Functional Analysis)

  • Walter Rudin’s “Principles of Mathematical Analysis” (Undergraduate Real Analysis)