Topics - Topics in Analysis Part III, Autumn 2010 T. W....

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Unformatted text preview: Topics in Analysis Part III, Autumn 2010 T. W. K¨orner September 22, 2010 Small print This is just a first draft for the course. The content of the course will be what I say, not what these notes say. Experience shows that skeleton notes (at least when I write them) are very error prone so use these notes with care. I should very much appreciate being told of any corrections or possible improvements and might even part with a small reward to the first finder of particular errors. Contents 1 Introduction 2 2 Baire category 3 3 Non-existence of functions of several variables 5 4 The principle of uniform boundedness 7 5 Countable choice and Baire’s theorem 11 6 Zorn’s lemma and Tychonov’s theorem 13 7 The Hahn-Banach theorem 18 8 Three more uses of Hahn-Banach 21 9 The Rivlin-Shapiro formula 25 10 Uniqueness of Fourier series 27 11 A first look at distributions 31 1 12 The support of distribution 35 13 Distributions on T 37 1 Introduction This course splits into two parts. The first part takes a look at the Baire category theorem, Tychonov’s theorem the Hahn Banach theorem together with some of their consequences. There will be two or three lectures of fairly abstract set theory but the the rest of the course is pretty concrete. The second half of the course will look at the theory of distributions. (The general approach is that of [3] but the main application is different.) I shall therefore assume that you know what is a normed space, and what is a a linear map and that you can do the following exercise. Exercise 1. Let ( X, bardbl bardbl X ) and ( Y, bardbl bardbl Y ) be normed spaces. (i) If T : X → Y is linear, then T is continuous if and only if there exists a constant K such that bardbl Tx bardbl Y ≤ K bardbl x bardbl X for all x ∈ X . (ii) If T : X → Y is linear and x ∈ X , then T is continuous at x if and only if there exists a constant K such that bardbl Tx bardbl Y ≤ K bardbl x bardbl X for all x ∈ X . (iii) If we write L ( X,Y ) for the space of continuous linear maps from X to Y and write bardbl T bardbl = sup {bardbl Tx bardbl Y : bardbl x bardbl X = 1 , x ∈ X } then ( L ( X,Y ) , bardbl bardbl ) is a normed space. I also assume familiarity with the concept of a metric space and a complete metric space. You should be able to do at least parts (i) and (ii) of the following exercise (part (iii) is a little harder). Exercise 2. Let ( X, bardbl bardbl X ) and ( Y, bardbl bardbl Y ) be normed spaces. (i) If ( Y, bardbl bardbl Y ) is complete then ( L ( X,Y ) , bardbl bardbl ) is. (ii) Consider the set s of sequences x = ( x 1 ,x 2 ,... ) in which only finitely many of the x j are non-zero. Explain briefly how s may be considered as a vector space....
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This note was uploaded on 02/18/2012 for the course MATH 533 taught by Professor Drewarmstrong during the Fall '11 term at University of Miami.

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Topics - Topics in Analysis Part III, Autumn 2010 T. W....

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