References

In my university studies I was very intrigued by this page.

Therefore I would like to keep a personal list of references that in the years I have found particularly useful and which I wished I had known sooner. I hope this list will spare some time to people which have similar interests.

This list is not to be considered as a complete set reference, but more as some books which I personally like very much and which I would strongly recommend.

Also this list is very much "work in progress".

Mathematics

Analysis

Lang, Serge. Complex analysis. Springer-Verlag, Forth edition, 2010.

This book is an introduction to complex analysis which is exceptionally clear and pleasant to read. I agree with Conway on the view of complex analysis as "an introduction to mathematics" (quote is from the preface of the book: Conway, John B. Functions of one complex variable. Vol. 1, Second edition, Springer-Verlag, 1978. At the moment I have not read this book, hence it is not included in this list, but it probably will be once I get around taking a look at it). Complex analysis in one variable should be learned as early as possible. Especially before learning about operators on Hilbert spaces.

Lang, Serge. Real and Functional Analysis. Springer-Verlag, 1993.

This is a great book. I particularly like the presentation of measure theory in this book. It is quite unique, clear and concise. This book includes a lot of information, among which: measure theory, real analysis, operators on Hilbert spaces, and also global analysis. Personally I think the proof of the spectral theorem for unbounded operators is unclear. But it is just one of the many proofs and it can be looked up somewhere else (see section on operators on Hilbert spaces).

Linear operators on Hilbert spaces

Kreyszig, Erwin. Introductory functional analysis with applications. John Wiley and sons, 1978.

This book looks long but it reads like a novel. It does not use measure theory but it is rigorous. Very good as a first book on the topic.

Weidmann, Joachim. Linear operators in Hilbert Spaces. Springer-Verlag, 1980.

This is my go to book on the topic. Perfect after a browsing Kreyszig.

Representation of Lie groups and Lie algebras

There are many good books. Here I mention only my very favorite ones. In my studies I was introduced first to the representation theory of Lie algebras with the book by Humphreys. Personally I found that approach very algebraic and not intuitive. Things started to make sense to me only after learning the "global picture" i.e. the representation theory of Lie groups. Then one also appreciate more the algebraic point of view. For example the notion of universal enveloping algebra can seen purely algebraically or as the algebra of left-invariant differential operators on a Lie group. I think the algebraic definition becomes clearer after one sees the differential one. This list is heavily influenced by this "global" point of view.

Barut, Raczka. Theory Of Group Representations And Applications. 1986, World Scientific.

This is one of my favorite books of all time. It has an incredible amount of information and it is rigorous, general, and easy to read. Really a masterpiece. My suggestion is to skip the first chapter (on Lie algebras) and go back to it later after reading Chapter 7 (representation of compact groups). Actually the first chapter is there only as a list of results without most proofs. One should look at other books like the Humphreys for the proofs. Chapter 7 is great and could very well be the first chapter to read.

Taylor, Michael. Lectures on Lie Groups. (2018)

These lecture notes used to be available online. The link on the author webpage seems broken at the moment. Hopefully they will become available again because I think they exceptional: clear, short, to the point, and with a lot of information that is often left out even in longer presentations.

Simon. Representations of finite and compact groups American Mathematical Society, 1996

Very clear on things that are often left out in other texts.

Warner, Garth. Harmonic Analysis on Semi-simple Lie Groups. Springer-Verlag, 1972. Two volumes.

This is a complete reference. Very useful for the details on non-compact groups where the analysis becomes crucial. I like very much the fact that is extremely complete but manages to be accessible by making every paragraph almost self-complete. Colossal amazing work.