Nota bene. My distinction between Pure and Applied books, and the categorization of books into subfields is almost completely arbitrary.
If you know how to construct the real line and have a feeling for least upper bound and greastest lower bound properties of a set, I think this is one of the best introductory books to analysis and topological spaces. In my opinion, Sutherland's introduction and treatment of compactness is very intuitive.
The beginning of this book was a little rough for me. I fell asleep a handful of times. I am a few chapters in now, and it is not so bad anymore. I will give a few more comments after a second reading.
I have only read the first chapter. I am not convinced by the organization of this book. I suspect it also needs at least a second read to full appreciate.
This is my favorite linear algebra book. The book sections are organized by very narrow topics and the exercises are nice. Unfortunately, if this is the first time you are learning linear algebra, this is definitely not the best choice of book.
I have not read the entirety of this book, but the first few chapters are amazing. I came across this book when I was trying to understand a computational method called HELM used for power networks.
I have only glanced at this book. But it is the standard text for an introduction to abstract algebra. Nothing to be said beyond that.
This is the standard introductory text book to analysis. Rudin does not have much exposition, but he is very efficient. In my experience, his efficiency is not the usual for most people doing research in the mathematical sciences, especially early in their careers. I appreciated his style, but I would not use it as a model for myself.
This book is currently by my bedside, and I have only finished Chapter I. It has really nice exposition. I am interested in seeing how this style plays out in the latter part of Chapter III, and Chapter IV.
I have only looked at parts of this book. It is the classical text in the subject.
This is the standard graduate text in analysis. I was not crazy about the proof of the Riesz Representation Theorem. I prefered Bass's way of first proving the Carathéodory extension theorem, and then proving the representation theorem. Also, the Carathéodory extension theorem is pretty useful in probability theory on its own.
Probability & Mathematical Statistics
This is definitely my favorite introductory book on measure theoretic probability. It is nice and modern, and the exercises are great.
This book is more or less a complete introduction to elementary and measure theoretic probability. I have not read it the whole way through, but I often use it as a reference.
I have only read specific chapters from this book, but it is my go to book for almost all theoretical statistics results. It baffles me how one person could have written such an extensive book.
I have not read this book in its entirety. I usually only pull it out when van der Vaart's book does not have what I need. I do intend on reading it though because I have had such a positive experience with it so far.
Here is where things get more exciting. You can tell on which side of the pure-applied continuum I fall. Also, you will notice that I have only one book on "machine learning" and "artificial intelligence". To me, these are engineering disciplines built on top of computer science, statistics, optimization, approximation theory, information theory and specific domain knowledge. As such, innovations in these fields are in many ways still justified using empirical evidence. Although the mathematical sciences are catching up, there are very few good texts on the subject.
General Applied Analysis
Lanczos is hugely influential. I have this book sitting on my shelf, but have yet to read it.
This is a very easy to read book. Great exposition. Great perspectives. Also, the entire thing is produced in matlab, and so there is an equal emphasis on approximation practice. It is a nice read, but this book shines if you work though the programming as well.
I have only read the first five chapters of this book so far, and it is not a very rigorous introduction to the subject. An understanding of generalized functions is necessary to fill in the gaps completely. However, the ideas are conveyed well.
This is a nice introduction to the topic, and, just as the title promises, it is indeed mathematical. I enjoyed the presentation in this book.
Data Analysis & Applied Statistics
I use this book mainly as a reference. I do intend to read it at some point, once I have my own copy.
I have used this book as a reference on a variety of topics. It is also a very nice read. McCullagh's sense of humor comes through in the exposition.
This was once described to me as the bible of Stanford's data analysis. Understandly, this book is very applied and discusses a number of modern methods applied to data analysis. My big complaint about this book is that it sweeps under the rug the problems of dependent examples. The world is never independent and identically distributed.
Optimization & Control
The first part of this book is a great introduction to convex analysis. I would really recommend it before studying any functional analysis topics. On the other hand, it does not present any algorithms.
This is also a theoretical book on nonlinear programming. Although there are no explicitly stated algorithms, they can be constructed from the exposition. This book also has one of the best discussions of duality. One disadvantage of this book is that it does not discuss trust region methods.
This is the book if you want to understand how to implement numerical optimization algorithms for differentiable objective functions, and to understand why they work. There is a great balance of theory and practice. However, in my opinion, the ordering of the chapters could be better, and the presentation of constrained programming could use some improvement.