Random Quote: Here is my advice to you, the adventurers – fear will show you the way; walk steadily toward it, for otherwise you will always be running. - Yossi Ghinsberg
My interests are in probability, functional analysis and statistics. At the moment, most of my research centers on numerical analysis, stochastic processes and simulation.
If you are interested in analysis and probability with a view to mathematical economics then here is my reading list. I am extremely fussy, and I think that all of these books are great.
The book by Bartle and Sherbert is where you should start. It’s still the best introduction to analysis that I know. Kolmogorov and Fomin is old but still a very nice read on functional analysis. Pollard has a great style and interesting topics. He uses a non-standard notation that takes a while to get your head around but feels like the right way to do probability once you get used to it. Aliprantis and Burkinshaw is an excellent graduate text with lots of exercises. Dudley is great but he frightens me a little bit. Schilling is beautiful and useful. Cheney is beautiful too, and a nice balance between theory and applications.
For background reading, try the wonderful book called “In Search of Infinity” by N. Ya. Vilenkin.
I love the work of Escher, especially this one.
Incidentally, the man generally recognized as the founder of modern functional analysis is Stefan Banach. Many fundamental results in normed linear space stem from his 1922 thesis. Another important contributor to the theory of abstract spaces was Maurice Frechet. Frechet also helped generalize the Lebesgue integral to operate on functions of arbitrary domain. Within the field of Markov processes, a bright and shining light was created by the short career of Wolfgang Doeblin. It’s been a huge privilege to understand and even contribute a tiny amount to his wonderful ideas.