Gow - Topics in Analysis T. W. K¨orner September 18, 2007...

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Unformatted text preview: Topics in Analysis T. W. K¨orner September 18, 2007 Small print This course is not intended as a foundation for further courses, so the lecturer is allowed substantial flexibility. Exam questions will be set on the course as given (so, if I do not have time to cover everything in these notes, the topics examined will be those actually lectured). Unless there is more time to spare than I expect, I will not talk about topological spaces but confine myself to ordinary Euclidean spaces R n , C and complete metric spaces. Some results will be marked with a ⋆ because most students will have met the ideas before. The proofs of these results will not be given in lectures but I intend to give additional classes and I will be happy to discuss these or anything else to do with the course during the classes. I have sketched solutions to the exercises which should be available for supervisors from the departmental secretaries or in tex, ps, pdf and dvi format by e-mail. I should very much appreciate being told of any corrections or possible improvements to these notes and might even part with a small reward to the first finder of particular errors. These notes are written in L A T E X2 ε and should be available in tex, ps, pdf and dvi format from my home page http://www.dpmms.cam.ac.uk/˜twk/ Contents 1 Metric spaces 2 2 Compact sets in Euclidean Space 5 3 Laplace’s equation 7 4 Fixed points 10 5 Non-zero sum games 12 6 Dividing the pot 14 7 Approximation by polynomials 16 1 8 Best approximation by polynomials 20 9 Gaussian integration 21 10 Distance and compact sets 23 11 Runge’s theorem 25 12 Odd numbers 29 13 The Baire category theorem 33 14 Continued fractions 35 15 Continued fractions (continued) 38 16 Winding numbers 44 17 Exercise Sheet 1 47 18 Exercise Sheet 2 50 19 Exercise Sheet 3 54 20 Exercise Sheet 4 60 1 Metric spaces This section is devoted to fairly formal preliminaries. Things get more inter- esting in the next section and the course gets fully under way in the third. Both those students who find the early material worryingly familiar and those who find it worryingly unfamiliar are asked to suspend judgement until then. Most Part II students will be familiar with the notion of a metric space. Definition 1.1. Suppose that X is a non-empty set and d : X 2 → R is a function such that (i) d ( x,y ) ≥ for all x, y ∈ X . (ii) d ( x,y ) = 0 if and only if x = y . (iii) d ( x,y ) = d ( y,x ) for all x, y ∈ X . (iv) d ( x,y ) + d ( y,z ) ≥ d ( x,z ) for all x, y, z ∈ X . (This is called the ‘triangle inequality’ after the result in Euclidean geometry that the sum of the lengths of two sides of a triangle is at least as great as the length of the third side.) 2 Then we say that d is a metric on X and that ( X,d ) is a metric space....
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