Part I (Michaelmas Term) of the yearlong module MAU22200: Advanced Analysis

School of Mathematics, Trinity College

Lecturer Dmitri Zaitsev

Organization and Content:

For additional up-to date information, please see the module page MAU22200-A-YEAR12-202021 ADVANCED ANALYSIS on TCD Blackboard.


To be posted regularly to Blackboard as videos with slides, starting from the first week (of Sept 28). I'll be sending respective annoucements through Blackboard at the beginning.


Weekly starting next week (Oct 5-9). Live online for now, until further notice, via Microsoft Teams. More details will be posted and announced via Blackboard.


Problem sheets will be posted in advance for each tutorial. Only some of the sheets will be marked (not the first 2 tutorials), to ease the burden of the online submission and marking, and contribute for 10% of your total mark. It is important to be able to do all the problems, including unmarked ones, to ensure you are prepared for the exam.

Public Discussion Forum on Gitlab:

An alternative to our Discussion Board on Blackboard. Please join to ask questions, give feedback or help others if you know answers. Unlike many other forums, here you can type mathematics formulas in Latex, and that feature will make it more convenient for us to give a better answer by including properly looking mathematical formulas. Gitlab is an important discussion/collaboration tool (not only for coding projects) that we are using as a forum. Also beneficial for strengthening TCD Graduate Attributes.

Module outline (Michaelmas Term):

Metric spaces: metric axioms, metrics d_p in R^n, discrete metric, open balls, bounded/open/closed sets, convergence of sequences, continuity and uniform continuity of functions, Lipschitz continuity, pointwise and uniform convergence of sequences of functions, Cauchy sequences, complete metric, completion, contractions.

Topological spaces: topology, metrisable topology, convergence, open/closed sets, closure, interior, boundary, neighbourhoods, limit point of a subset, continuity, subspaces, product topology, homeomorphisms, Hausdorff topology, connectedness and compactness.

Normed vector spaces: norm, bounded linear operator, operator norm, Euclidean norm, equivalent norms, Banach space, absolute convergence, invertible linear operator, dual space.


W.A. Sutherland, Introduction to Metric and Topological spaces, Oxford University Press, 1975, or the 2nd edition from 2009.
G. F. Simmons, Introduction to topology and modern analysis, McGraw Hill Book Co., 1963.
E.T. Copson, Metric spaces, Cambridge University Press, 1968.
A.N. Kolmogorov and S.V. Fomin, Elements of the theory of functions and functional analysis Vol. 1, Graylock Press, 1957.
W. Rudin, Principles of mathematical analysis, McGraw-Hill Book Co., 1976.
K. Kuratowski, Introduction to set theory and topology, Pergamon Press, 1972.
C. W. Patty, Foundations of topology, PWS-KENT Publishing Co., Boston, MA, 1993.
J. Dieudonné, Foundations of modern analysis., Academic Press, 1969 or the newer 2011 edition.

Past related modules:

MAU22200 - Advanced Analysis (Metric Spaces) 2019 by John Stalker with Problem Sheets and Solutions.
MA2223: Metric Spaces 2018 by Sergey Mozgovoy with Lecture Notes and Problem Sheets.
MA2223: Metric Spaces 2017 by Paschalis Karageorgis with Lecture Notes, Problem Sheets and Solutions.
MA2223: Metric Spaces 2016 by Sergey Mozgovoy with Lecture Notes and Problem Sheets.
MA2223: Metric Spaces 2014 by Sergey Mozgovoy with Lecture Notes and Problem Sheets.
MA2223: Metric Spaces 2013 by Sergey Mozgovoy with Lecture Notes and Problem Sheets.
MA2223: Metric Spaces 2012 by Sergey Mozgovoy with Problem Sheets.
212 - Metric Spaces and Topology 2004-05 by myself with Problem Sheets.
212 (Topology) 1989-2001 by David Wilkins with Lecture Notes and Problem Sheets.

Links to broaden your horizon:

Honors Analysis by Terence Tao (2006 Fields Medal)
How should mathematics be taught to non-mathematicians? by Timothy Gowers (1998 Fields Medal)
Why Do We Learn Math? by Better Explained
The Crowdsourced Guide to Learning maintained by the online learning platform FutureLearn

Past examinations:

TCD examination papers (2012 - present)
TCD examination papers (1998 - 2012)
School of Mathematics examination papers (1992 - 2000) with Course Notes and Problem Sheets.


I will appreciate any (also critical) suggestions that you may have for the module. Let me know your opinion, what can/should be improved, avoided etc. and I'll try my best to take it into account.