Linear Functional Analysis

aka “Advanced analysis: Linear Analysis”, “Introduction to Functional Analysis”, “Banach and Hilbert spaces”

Autumn Semester 2007

I hope to learn Blackboard so soon, that this page becomes redundant. In the meantime, here are:

You will get one set every two weeks, and the deadline for handing in your work is two weeks later as well. Your examination grade will be based on this homework.

Here is an outline of the course as it was given a couple of years ago. I will closely follow this scheme. I am lecturing for the first seven weeks. Then we have the “breekweek”. After that, my colleague Marcel de Jeu takes over. The first week after the “breekweek”, however, he will be absent. I will need the extra lecture in order to finish at the right point. Marcel will also need to grab an extra week at the very end of the course, to compensate for his missed lecutre, and to arrive at where he is headed to. So you lucky people get, completely free, one extra lecture.

As I wrote those words (20 September: the last day of Summer), I had just given my third lecture, and just told you that the second set of exercises was posted on this site. I was running just slightly behind the outline, but so far I had covered essentially the same material; the accent and style was just a little bit different.

Here are outlines of the lectures given so far (slightly edited text taken from the old outline):
Here are slides which I used in the lectures so far:

General comments
The course is intended to provide preparation for those wishing to follow the Masters level course on functional analysis, which already assumes basic knowledge of the field. At the same time, this basic knowledge is basic knowledge for anyone seriously interested in stochastics, analysis or operations research. But in any case: EVERY mathematician MUST know what is a Banach space, what is a Hilbert space, what are the other basic concepts and first main results in the field.

The course is based on the book by Bryan Ryngge and Martin Youngson: Linear Functional Analysis (Springer). Get it and start reading it. In the first two weeks we will cover the first chapter - an overview of the requisite prior knowledge - the basic elements of linear algebra, theory of metric spaces, and the Lebesgue integral.

Since many students did not meet that integral before, the second week's lecture will simply be a crash-course in the Lebesgue integral: the right integral for our purposes. It simply extends the Riemann integral which you already know and love, in a totally natural way, to a larger class of functions. With it in our tool-kit, we mathematicians can integrate the largest possible class of functions, without losing any good properties of the old integral, at the same time adding new good properties, in particular, concerning interchange of the operations of integration and taking limits. Hopefully you will be enticed to learn this subject in depth and detail, in another course. (And if you want to study statistics and probability, you will have to, anyway.)

Henri Lebesgue, David Hilbert, and Stefan Banach were giants of the mathematics of the 20th century. Their work brought analysis to a new higher level from which it has had enormous impact in physics, stochastics, engineering, finance, ... in short, on science as we know it. Mathematical abstraction helped mathematicians to get the final answers to long standing applied problems. For instance, the development of Fourier theory from Fourier himself led Riemann to invent his integral, then Lebesgue to generalise it, allowing us to make a final and elegant description of which functions possess Fourier series. Read about their lives at the Saint Andrews site on the history of mathematics. It makes fascinating reading.

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