# MA412: Functional Analysis and its Applications 2009/10

## General information about MA412: Functional
Analysis and its Applications

### Lectures and Classes - Timetable

This is a half-unit course, with lectures in the **Michaelmas Term**.

Classes will start in Week 2 of the Michaelmas Term.

The full timetable for the course can be found on the Timetables website:
http://www.lse.ac.uk/admin/timetables/confirmed/module_sessional/ma/28.htm

### Exercises

Homeworks will be set weekly. The homework
has to be handed in by 5:00 p.m. on Mondays. No late work will be
accepted. If you don't finish the assignment, hand in whatever you
have. Label all work with your name and the course number (MA412).
Corrected work will be handed back and discussed in the
classes. Solutions for some of the homework exercises will be given
out as well. This means that not all exercises need to be discussed in
class.
### Course material

A complete set of lecture notes will be provided at the start of the
term. Additionally, the student may wish to refer to some of the books
mentioned in the bibliography of the lecture notes.
Among others, the following texts may be used for supplementary reading.
1. E. Kreyszig.
* Introductory Functional Analysis with Applications.*
Wiley, 1989.

2. D.G. Luenberger.
* Optimization by Vector Space Methods.*
Wiley, 1969.

3. W. Rudin.
* Real and Complex Analysis; 3rd edition. *
McGraw Hill, 1987.

### Lectures

During the lectures, the theory will be developed and explained, proofs
given, and many examples demonstrated.

In the Summer Term there will be additional lectures, mainly for revision
purposes.
### Classes and Exercises

In this course, as in other courses in Mathematics, it is very important
that all homework questions are attempted and handed in for grading. It
is vital to get practice in the various techniques covered in the
course.
It is also important to hand in homework, so that feedback on it can be
given. Corrected work will be handed back and discussed in the class.
### Office Hours

The office hours are meant for any questions and problems with the course
material that have not or cannot be covered in the normal lectures and
classes. You are strongly recommended to make use of them.

The times of the office hours for all lecturers and class teachers can be found
on the departmental
office hours page.

## Course description of MA412: Functional
Analysis and its Applications

### Overview

This course aims at familiarizing the student with the basic concepts,
principles and methods of functional analysis and its applications.
Functional analysis plays an important role in the applied sciences as
well as in mathematics itself.
Roughly speaking, functional analysis develops the tools from calculus
and linear algebra further to the more general setting where one has
vector spaces comprising functions or general abstract
infinite-dimensional vector spaces. Problems from various application
areas can then be conveniently posed in this common general set up,
and solved using the techniques of functional analysis.
The basic objects studied in functional analysis are vector spaces
with a notion of distance between vectors, and continuous maps
between such vector spaces. This interplay between the algebraic
and analytic setting gives rise to many interesting and useful
results, which have a wide range of applicability to diverse
mathematical problems, such as from numerical analysis,
differential and integral equations, optimization and approximation theory.
### Course Content

The course will cover the following topics:

** 1. Normed linear spaces and Banach spaces. **

** 2. Continuity and bounded linear operators. **

** 3. Differentiation.**

** 4. Geometry of inner product spaces. Hilbert spaces. **

** 5. Compact operators. **

## Assessment and Exam Papers

The assessment for this course will be based on two parts:

One set of coursework will count for 10%. This coursework will be set
in the middle of the term, and has to be handed in within
2 weeks.

The remaining 90% will be based on a formal 2-hour examination in the
Summer Term.
## Past exam papers for this course

**Please note: **students are advised not to rely too heavily on past exam
papers when revising for their exams, as they can only offer a limited
indication of what might be covered in a future exam. For further information,
please see the guidance here:
http://www.maths.lse.ac.uk/examinations_in_mathematics.html#past_papers

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School of Economics & Political Science
2009 Last changed: 14th October 2009

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