## MAS6352 Analysis II

Both semesters, 2017/18 | 20 Credits | ||||

Lecturer: | Prof David Applebaum | uses MOLE | |||

Aims | 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
- 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.
- 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.

**Semester 2:**

## Aims

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 chosen from five, 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

**R**

^{n}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 x

_{n}→ x with d(x

_{n},x)→0 in

**R**. Convergence in

**R**

^{2}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

**R**

^{n}and function spaces.

**6. Iteration and Contraction**

(4 lectures) Iteration as a method to solve problems. Examples in

**R**,

**R**

^{2}. 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.