MAS6352A Analysis IIA

 Semester 1, 2021/22 5 Credits Lecturer: Dr Dimitrios Roxanas Reading List Aims Outcomes Teaching Methods Assessment Full Syllabus

When working on the real line, there is only one standard concept of convergence. This course studies convergence in the more general setting of a metric space, particularly the important cases of Rk and function spaces, such as the space of continuous functions. A metric space is a set with a "distance function" which is governed by just three simple rules, from which the entire body of analysis follows. The course follows on from courses in basic Analysis in R, and adapts some of the ideas to the more general setting. The course includes the Contraction Mapping Theorem, which guarantees the convergence of quite general iterative processes; there are applications to many other areas of mathematics, such as to the solvability of differential equations.

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

Outline syllabus

• Metric spaces: definition, properties and examples
• Convergence of sequences
• Open and closed subsets
• Continuity
• Cauchy sequences, completeness
• Compactness
• Iteration and the Contraction Mapping 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 provide a basic course in Analysis in this setting
• To reinforce ideas of proof
• To illustrate the power of abstraction and show why it is worthwhile
• To provide a foundation for later Analysis courses

Learning outcomes

• decide whether sequences converge, and find the limits in a number of concrete and abstract spaces;
• decide whether given functions are or are not metrics;
• determine, in a variety of contexts, whether a given subset of a metric space is closed;
• discover whether examples of metric spaces are complete using Cauchy sequences;
• understand iterative processes on general spaces;
• understand compactness;
• prove and apply the Contraction Mapping Theorem.

Teaching methods

In addition to lectures and office hours, we will have an (optional) tutorial every few weeks, on Thursday 16.00-17.00. Dates for the tutorials: 14/10 (Hicks LTA), 28/10 (Hicks LTA), 18/11 (Hicks LT**1**), 2/12 (Hicks LT**1**), 9/12 (Hicks LTA). - Please keep an eye out for announcements about room changes.

Office Hours: Monday 1-2pm (on Blackboard, there is a link on the left-hand-side panel) and by appointment.
You will be given 3 non-assessed homeworks (marked and returned for feedback). They will be released (on Crowdmark) on 6/10, 27/10, and 24/11 respectively. These dates are tentative.

20 lectures, no tutorials

Assessment

2.5 hour-long exam worth 75%. Additional project worth 25%.

Full syllabus

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. 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 Heineâ€“Borel property.
7. 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.