George Coote's Blog

Book recommendations

This is a list of books I have quite enjoyed and plan to read more of.

Functional Analysis

Introductory/GENERAL

These all should be accessible with some knowledge of complex analysis, linear algebra, general topology, and general “analytic familiarity”.

  • Linear Functional Analysis by Rynne and Youngson – I think this was the first functional analysis book I used and was my favoured book when starting out. It doesn’t go very deep – largely just covering Functional Analysis I at Warwick (and probably less than Part II Linear Analysis), but I remember it being very clear and accessible.
  • An Introduction to Functional Analysis by Robinson – covers the two third-year functional analysis courses at Warwick. Done in the setting of normed vector spaces and inner product spaces, pretty much everything you would want from a first exposure to functional analysis. Very clear, no complaints.
  • Functional Analysis (“Grandpa Rudin”) by Rudin – introduces theorems that you will already know (e.g. Open Mapping Theorem, Principle of Uniform Boundedness) in more general settings. Presentation starts at topological vector spaces, (vector spaces with a, usually Hausdorff, topology that makes vector multiplication and scalar multiplication continuous – normed vector spaces are of course examples) almost as general as it gets really, and gradually specialises. Would recommend as a second course unless you are fairly confident and/or want to learn functional analysis rapidly. Contains introduction to unbounded operators and the spectral theorem for normal operators.
  • A Course in Functional Analysis by Conway – A good reference, used this to read about the GNS construction for the first time and it was very well-written and understandable.
  • Methods of Modern Mathematical Physics I by Reed and Simon – this goes from scratch again though I never used it as a first exposure to functional analysis. I have mainly used it for its content on unbounded operators and the spectral theorem, though I intend to explore Chapter V in the future.
  • Mathematical Methods in Quantum Mechanics With Applications to Schrodinger Operators by Teschl
BANACH SPACES
  • Functional Analysis and Infinite-Dimensional Geometry by various – in principle this should be accessible as a first course, though I would not recommend it. The first part of the book is an excellent complement to the first half of the Part III Functional Analysis course, especially the exercises, and I have enjoyed reading bits of topics beyond that. The focus is on Banach spaces with some content on locally convex spaces.
SPECTRAL THEORY
  • Introduction to Spectral Theory by Hislop and Segal – used this to learn about Schrodinger operators for the first time, and for that it was pretty solid. It covers the basic theorems that you would want (e.g. the min-max principle for semibounded-below operators, self-adjointness criteria)
  • Spectral Theory and Differential Operators by Edmunds and Evans – used this to learn about the essential spectrum and Fredholm operators. It has a lot on PDEs which I am not particularly interested in but I expect this to be a useful source going forward. There are many typos that I found, even as much as one trivial typographical error every few pages. Still thought it was good. Not to be confused with the book of the same title by Davies.
banach algebras and c*-algebras

Others

COMPUTABILITY THEORY
  • Turing Computability by Soare – very accessible introduction to aspects of Turing computability and the arithmetical hierarchy. Very well-written, usually clear and always concise.
  • Andrew Marks’ Computability Theory notes – granted, not a book, but I found this useful for learning about Godel’s incompleteness theorem and non-standard models of the natural numbers. Also has an introduction to the arithmetical hierarchy.
SET THEORY
ANALYTIC NUMBER THEORY