## MAS6352 Analysis II

 Both semesters, 2021/22 20 Credits Lecturer: Dr Nic Freeman Timetable Aims Outcomes Assessment Full Syllabus

The module will give students an additional opportunity to develop skills in modern analysis as well as providing a rigorous foundation for probability theory. In particular it would form a companion course to MAS6340 (Analysis I) and MAS6052 (Stochastic Processes and Finance), the latter of which is fundamentally dependent on measure-theoretic ideas. In the first semester, ideas of convergence of iterative processes are explored in the more general framework of metric spaces. A metric space is a set with a "distance function" which is governed by just three simple rules, from which the entire analysis follows. Semester 2 studies "measure theory", a branch of mathematics which evolves from the idea of "weighing" a set by attaching a non-negative number to it which signifies its worth. This generalises the usual physical ideas of length, area and mass as well as probability. It turns out (as we will see in the course) that these ideas are vital for developing the modern theory of integration.

There are no prerequisites for this module.
No other modules have this module as a prerequisite.

## Outline syllabus

Semester 1:
• Metric spaces: definition, properties and examples
• Convergence of sequences
• Open and closed subsets
• Continuity
• Cauchy sequences, completeness
• Iteration and the Contraction Mapping Theorem
• Compactness
Semester 2:
• The scope of measure theory,
• σ-algebras,
• Properties of measures,
• Measurable functions,
• The Lebesgue integral,
• Interchange of limit and integral,
• Probability from a measure theoretic viewpoint,
• Characteristic functions,
• The central limit theorem.

## Aims

• To point out that iterative processes and convergence of sequences occur in many areas of mathematics, and to develop a general context in which to study these processes
• To reinforce ideas of proof
• To illustrate the power of abstraction and show why it is worthwhile
• give a more rigorous introduction to the theory of measure.
• develop the ideas of Lebesgue integration and its properties.
• recall the concepts of probability theory and consider them from a measure theoretic point of view.
• prove the Strong Law of Large Numbers and the Central Limit Theorem using these methods.

## Learning outcomes

By the end of the unit, a candidate will be able to demonstrate the ability to, the capability for... 1. understand iterative processes on general spaces; 2. decide whether sequences converge, and find the limits in a number of concrete and abstract spaces; 3. decide whether given functions are or are not metrics; 4. determine, in a variety of contexts, whether a given subset of a metric space is closed; 5. discover whether examples of metric spaces are complete using Cauchy sequences; 6. prove and apply the Contraction Mapping Theorem; 7. understand compactness. 8. understand why a more sophisticated theory of integration and measure is needed; 9. show that certain functions are measurable; 10. construct the Lebesgue integral; 11. understand properties of the Lebesgue integral; 12. develop probabilistic concepts (random variables, expectation and limits) within the framework of measure theory. Over and above these level 3 outcomes, students on this level 6 course will 13. understand basic concepts of topological spaces; 14. show a wider understanding and appreciation of the role of metric spaces in mathematics; 15. show a wider understanding of some aspects of measures that are dealt with less fully on the level 3 course, such as product measures and Fubini #39;s theorem.
40 lectures, no tutorials

## Assessment

The module will be assessed by formal, closed book examinations at the end of each semester.

The Semester 1 exam will be a 2.5 hour written exam with four questions all of which are compulsory, worth 75
The Semester 2 exam will be a 2.5 hour written exam with a choice of three questions from 4.

## Full syllabus

Semester 1

1. Metric spaces
(4 lectures)
Distance functions. Definition of metric space. Review of suprema and infima. Examples including Rn and function spaces. Some basic properties of metrics. Subspaces. Closed balls and open balls.
2. Convergence of sequences
(3 lectures)
Definition in terms of N and ϵ. Examples. Basic properties: uniqueness of limit, equivalence of xn→ x with d(xn,x)→0 in R. Convergence in R2 means convergence componentwise. Convergence in function spaces and pointwise convergence.
3. Closed and open sets
(2 lectures)
Definition of closed sets and open sets. Behaviour under intersections and unions. Examples.
4. Continuity
(2 lectures)
Continuity in terms of sequences and in terms of ϵ and δ. Relation with closed sets and open sets. Examples.
5. Cauchy sequences and completeness
(2 lectures)
`Internal' tests for convergence, without knowledge of the limit. Cauchy sequences. Completeness. Bolzano-Weierstrass. Examples including Rn and function spaces.
6. Iteration and Contraction
(4 lectures)
Iteration as a method to solve problems. Examples in R, R2. Fixed points of iterations. Discussion of what should be required to guarantee convergence of iterative procedures to ﬁxed points. Contractions. Examples. The Contraction Mapping Principle. An application to linear algebra. A diﬀerential criterion for a function to be a contraction. Functions with the property that repeated application gives a contraction. Application to existence of solution of diﬀerential equations. Examples.
7. Compactness
(3 lectures)
Definition using subsequences. Compactness and continuity. Examples. Compact subsets of Euclidean spaces. Equivalent formulations in terms of total boundedness and completeness, and in terms of the Heine–Borel property.