MC240 Abstract Analysis

Next: MC241 Linear Algebra Up: Year 2 Previous: MC226 Numerical Linear Algebra

MC240 Abstract Analysis

Credits: 10

Convenor: Will Light

Semester: 2

Prerequisites:	essential: MC144, MC145, MC146, MC147, MC248, MC241
Assessment:	Continual assessment: 20%	One and a half hour exam: 80%

Lectures:	18	Classes:	none
Tutorials:	6	Private Study:	51
Labs:	none	Seminars:	none
Project:	none	Other:	none
Total:	75

Explanation of Pre-requisites

The course is essentially analytical in nature, so the algebraic prerequisites bear only tangentially on the course. Of major importance are MC146 and MC248.

Course Description

The aim of the course is to understand in an abstract way the two notions of continuity and integrability. At the same time, I want to convey my own enthusiasm for these two central achievements of 20th century mathematics!

Let's start with continuity. Suppose we have a function f mapping some sort of objects (which we will call `points') to numbers. What does it mean for the function to be continuous at the point p? Well, roughly speaking, if q is close to p, then f(q) must be close to f(p). OK so far? Well, not really. We've already been much too loose in the way we have formulated the problem. First off, we described the things on which f acts as `objects', and then used the more emotive word `points'. Then we went on to talk about these `objects' or `points' being close together. This places some restriction on the sort of objects allowed. For example, our set of objects could consist of all cheeses currently lying on the shelves of Sainsbury's supermarket. For a particular cheese p, f(p) (which remember has to be a real number) could be the weight of the cheese. Now I think you're going to have a hard job coming up with an appropriate definition of two cheeses being close together! Note here that the problem in determining a sensible meaning for `closeness' only occurs in the input space, or the domain of the function - the set of all cheeses in Sainsbury's. Closeness in the output space is easy to define isn't it? We would say that f(p) is close to f(q) if |f(p)-f(q)| is small. But hang on a minute. Even this idea is not precise enough for a mathematician (or even for an applied scientist, or person working is some sort of technological industry). If I was to discover that the north wall of my house was 2mm higher than the south wall, I wouldn't exactly be rushing to sue the builder for negligence. On the other hand, if the garage sets the gap between my spark plug electrodes with an error of 2mm, then I would be bleating in their ears that something is wrong with my car. So the notion of closeness needs to be expressed a lot more carefully. (The formalisation of the concept of closeness in $\hbox{I}\kern-.2em\hbox{R}$ leads inexorably via MC146 to the formal definition of continuity using $\epsilon$ and $\delta$ technology.) But around the turn of the century, mathematicians wanted to get away from the idea of `closeness'. This desire was bound up with another very sophisticated development in thinking, which has today become second nature to professional mathematicians. Traditionally, functions used to be things which mapped numbers to numbers, or more generally, `points' in $\hbox{I}\kern-.2em\hbox{R}^n$ to `points' in $\hbox{I}\kern-.2em\hbox{R}^m$ . But it became increasingly common to think of functions in a much more abstract way. In the vanguard of this development was the idea that the so-called `points' could themselves be functions. For example, the `points' could consist of all real-valued functions differentiable on the whole real line. Each `point' is now a function from $\hbox{I}\kern-.2em\hbox{R}$ to $\hbox{I}\kern-.2em\hbox{R}$ . Thus the function $f:\hbox{I}\kern-.2em\hbox{R}\rightarrow \hbox{I}\kern-.2em\hbox{R}$ defined by the rule f(x)=x² would be a `point'. Now consider the mapping called `differentiation'. Applying this mapping to the `point' f gives the new function $g:\hbox{I}\kern-.2em\hbox{R}\rightarrow \hbox{I}\kern-.2em\hbox{R}$ defined by the rule g(x)=2x. Mathematicians wanted to study such types of mappings or functions acting on `points' which are themselves very sophisticated objects. A major force in this movement was the French mathematician Maurice Fréchet (1878-1973). The work of Fréchet and his contemporaries has had a profound effect on the way we teach you! In MC144, you first met the idea of a function. It probably seemed then pretty weird, because all you were given were two abstract sets A and B and the rule for getting from A to B, often called f. Nobody encouraged you to believe that either A or B were made up of numbers. Instead, we tried to convince you that A and B could be pretty much anything you liked: cheeses from Sainsbury's, functions, points in $\hbox{I}\kern-.2em\hbox{R}^n$ . However, because we are sensitive to the difficulty of this abstraction, we often used examples in which A and B were both (subsets of) $\hbox{I}\kern-.2em\hbox{R}$ . OK, let's hope you see that working with pretty abstract objects is part of modern mathematics, and let's return to our idea of continuity. My previous arguments have been designed to show that we need some idea of `closeness' before we can begin to make sense of continuity. In the example I have already introduced, where the `points' were themselves functions from $\hbox{I}\kern-.2em\hbox{R}$ to $\hbox{I}\kern-.2em\hbox{R}$ , it is not too difficult to come up with a sensible notion of when two `points' (functions) are close. MC248 takes this route, but we are bound for higher things! The fundamental question we shall answer in the first lecture of the course is

Continuity Challenge if A and B are abstract sets and f is a function with domain A and range B, what sort of structure is needed on A and B in order to be able to talk about continuity?

Of course, we cannot in this course be anarchists, and develop our own notion which fails to coincide with the MC146 notion of continuity. So there are constraints.

Let's go on now to talk about the other great concept of integration. At the outset, the `A'-level student knows that integration has two meanings - definite integration (which is the area under the curve), and indefinite integration (which is the reverse process to differentiation). Of course, the two are linked through the fundamental theorem of calculus. On a personal note, it's one of the great surprises to me, and something that I find absolutely fascinating, that the area under a curve f between a and b can be evaluated if only you know a function g such that g'=f. One version of the fundamental theorem is then that

$\begin{displaymath} \int_a^b f(x)\, dx = g(b)-g(a). \end{displaymath}$

Magic! But in this course, integration means definite integration, that is something to do with area under curves. Have you noticed we have switched back to functions $f:\hbox{I}\kern-.2em\hbox{R}\rightarrow \hbox{I}\kern-.2em\hbox{R}$ ? Without this switch the above discussion would not make sense. Also at this stage in your undergraduate career, all you have experienced is the Riemann integral. The upper and lower sums idea was introduced in a paper by Riemann in 1854. However, it soon became plain that Riemann's definition had some flaws. The work of Cantor (1874) had lead to the concept of the size of a set. Two sets have the same size if a 1-1 correspondence exists between them. Particular importance is attached to sets with the same size as the integers. Such sets are called countable. Cantor showed that the rational numbers are countable but the real numbers are not. Thus there are `as many' rationals as integers, but there are `far more' reals than integers.) What frustrated practitioners of the art of integration was that the function f which is zero at each rational value and 1 at all other values is, according to Cantor, 1 `most of the time'. It therefore ought to have $\int_a^bf(x)\, dx=\int_a^b 1\, dx = b-a$ for all values $a,b\in\hbox{I}\kern-.2em\hbox{R}$ . (By the way, in olden days the idea of integrating a function which had an absolutely huge (i.e., infinite) set of discontinuities was a major pastime.) What frustrated the masters of integration was that according to Riemann's definition, this function was not integrable. (It's not too hard to show this - look up the definition from MC248 and have a go!) So the Riemann integral had flaws, which were not really resolved until the thesis of Henri Lebesgue (1875-1941). Following Lebesgue, much the same programme was put in place for integration as for continuity:

Integration Challenge if A is an abstract set and f is a function with domain A and range in $\hbox{I}\kern-.2em\hbox{R}$ , what sort of structure is needed on A in order to be able to talk about integration?

The astute reader will pick up on a significant difference between the first question about continuity and the second about integrability. In our continuity challenge, we allowed both the domain and the range of the function to be arbitrary. In our integration challenge, we allow the domain to be arbitrary but restrict the range to be in the friendly old real numbers $\hbox{I}\kern-.2em\hbox{R}$ . It is in the nature of things that integration requires this, after all, we want $\int f$ to be a real number!

Aims

To convey the ideas of a topology, and the fundamental notions of continuity and compactness, along with the concept of a measure space, a measurable function and the concept of integration.

Objectives

Students will be able to define a topology, and say what is meant by a continuous function in a topological sense. They will be have the notion of a compact set, and the basic theorem about the attainment of sup and inf for a real-valued function. They will be able to define a measure space, a measurable function, and the integral of simple functions and measurable functions, where appropriate. They will have seen a selection of the results about the behaviour of the Lebesgue integral under limit processes.

Transferable Skills

The ability to present and understand arguments in an abstract setting.
The ability to apply taught concepts to new situations.

Syllabus

Definition of a topology. Definition of a $\sigma$ -algebra. Definition of a continuous function. Definition of a measurable function. Definition of continuity at a point in terms of neighbourhoods. Comments on the definitions. Continuity is equivalent to continuity at every point.
Conditions for $g\circ f$ to be continuous/measurable. u, v measurable implies $\Phi(u(x),v(x))$ measurable. f+g measurable, fg measurable, $\chi_E$ measurable.
smallest $\sigma$ -algebra containing a collection. How to construct a measure on a topological space, Lebesgue measure on $\hbox{I}\kern-.2em\hbox{R}$ . $f^{-1}\big((-\alpha,\infty]\big)$ measurable for all $\alpha$ implies f measurable.
closed sets in a topological space, limit points, set is closed if it contains its limit points, closure of a set, closure as smallest closed set containing given set, set closed iff it is same as its closure.
subspace topology, f|A continuous if f continuous, quotient spaces and the associated topology.
Compactness, continuous image of compact space is compact, closed subset of compact space is compact.
Hausdorff spaces, homemorphisms, compact subset of a Hausdorff space is closed.
Heine-Borel theorem
f 1-1 continuous onto from compact to Hausdorff is homeomorphism, Infinite subset of compact set must have a limit point.
Set in $\hbox{I}\kern-.2em\hbox{R}^n$ compact iff closed and bounded. Real-valued function attains its bounds on compact set.
Peano's space filling curve.
sup and lim sup are measurable. maxf,g is measurable. lim of every pw convergent sequence of measurable functions is measurable.
simple functions. Exists $\{s_n\}$ such that $s_n\rightarrow f$ for all measurable f. Definition of a measure and measure space. properties of measures with respect to countable unions, nested unions and intersections.
Integration of positive function via simple functions, elementary properties of the integral. $\int(f+g) = \int f + \int g$ and $\int_E s$ defines a measure.
Lebesgue monotone convergence theorem.
Fatou's lemma
The spaces $L^p(\mu)$ .

Reading list

Essential:

W. Rudin, Real and Complex Analysis, McGraw Hill, 1970.

Background:

W. Light, Introduction to Abstract Analysis, Chapman and Hall, 1990.

Details of Assessment

The coursework for the continual assessment consists of about 6 pieces of written work, consisting of problems from the problem sheets.

The written June examination lasts for one and one half hours, and contains 4 questions. Students must answer three.

Next: MC241 Linear Algebra Up: Year 2 Previous: MC226 Numerical Linear Algebra

Roy L. Crole
10/22/1998