I have created several problem sets to practice for Oxford's final examinations. I am developing this page and will update it shortly. Some of these problems are checked by my friend, Guoxi Liu, who is also an undergrad maths student at Oxford. Special thanks to him. Feel free to email me if there are any mistakes (I believe there are definitely a lot). These problem sets were set to be harder than the actual Oxford exam, so please don't panic if you get stuck on them!
Update 06.01.26. I have updated all mathematical errors in the functional analysis problems. I don't know when I will update a solution booklet for this, but I have all the solutions in mind.
Update 06.01.26. I have updated solution to problem 1 in the functional analysis problems.
Problems 1-7: B4.1; Problems 8-13: B4.2
For other resources, please see also Cambridge Past Papers, Linear Analysis, Topics in Analysis (not very helpful, easy problems regarding BCT) & Analysis of Functions (not very helpful also, most distribution/sobolev/PDE stuff). Also, C4.1 Further Functional Analysis past papers and Cambridge Part III Exam Paper (Functional Analysis) help.
Remark. I found no functional analysis exam harder than Oxford's.
Remark. It is interesting that, in part (c)(i) of problem 12, I originally wrote:
"Let K(X) be the space of compact operators on X. Show that K(X)\cap B_f(X) is dense in K(X)."
Then, I found that I could neither solve it nor give a counterexample in the case where X is a general Banach space. However, when X is a Hilbert space, the solution is short.
Please definitely email me if you could solve the case where X is a general Banach space.
For B8.1, Probability Theory and Examples by Rick Durrett is a very good resource. Check also Cambridge Part III Exam Paper, Advanced Probability. Part II Probability and Measures may also help with the Borel-Cantelli part of the course.
For B8.2, I would recommend the classic Continuous Martingales and Brownian Motion by Daniel Revuz and Marc Yor. This book is hard. Check also Cambridge Part III Exam Paper, Advanced Probability & Stochastic Calculus and Applications.
Measure Theory, Probability, and Stochastic Processes by Jean-François Le Gall is also a classic and hard book. It helps a lot with B8.1 and the first few parts of B8.2.
Problems (Complex Analysis Only)
Remark. Some problems are from UMichigan's Math PhD Qualifying Exam, Complex Analysis papers. They are quite hard.
Complex Analysis by Elias M. Stein and Rami Shakarchi has a lot of great problems. Check also Cambridge Past Papers, Part IB Analysis II/Analysis and Topology, Complex Analysis & Complex Variables.
You could also read more about entire functions, Euler products, Hadmard factorizations, Jensen's theorem, the proof of the Riemann mapping theorem, and conformal mappings; they are merely applications of theories presented in CA. Ch 5 and 8 of Stein are good resources.
If you want to get a very good understanding of residue calculus, you could study the Riemann zeta function (Ch 6-7 of Stein) and derive some explicit formulas.
Check UMichigan's Math PhD Qualifying Exam, Real Analysis papers. They are quite hard.
Real Analysis by Elias M. Stein and Rami Shakarchi has a lot of great problems.