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FA3 - Introduction to Functional Analysis Part III Autumn...

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Introduction to Functional Analysis Part III, Autumn 2004 T. W. K¨ orner October 21, 2004 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 Some notes on prerequisites 2 2 Baire category 4 3 Non-existence of functions of several variables 5 4 The principle of uniform boundedness 7 5 Zorn’s lemma and Tychonov’s theorem 11 6 The Hahn-Banach theorem 15 7 Banach algebras 17 8 Maximal ideals 21 9 Analytic functions 21 10 Maximal ideals 23 11 The Gelfand representation 24 1
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12 Finding the Gelfand representation 26 13 Three more uses of Hahn-Banach 29 14 The Rivlin-Shapiro formula 31 1 Some notes on prerequisites Many years ago it was more or less clear what could and what could not be assumed in an introductory functional analysis course. Since then, however, many of the concepts have drifted into courses at lower levels. 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, k k X ) and ( Y, k k 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 k Tx k Y K k x k X for all x X . (ii) If T : X Y is linear and x 0 X , then T is continuous at x 0 if and only if there exists a constant K such that k Tx k Y K k x k 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 k T k = sup {k Tx k Y : k x k X = 1 , x X } then ( L ( X, Y ) , k k ) 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, k k X ) and ( Y, k k Y ) be normed spaces. (i) If ( Y, k k Y ) is complete then ( L ( X, Y ) , k k ) 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. If we write k x k = sup j | x j | 2
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show that ( s, k k ) is a normed vector space which is not complete. (iii) If ( X, k k X ) is complete does it follow that ( L ( X, Y ) , k k ) is? Give a proof or a counter-example. The reader will notice that I have not distinguished between vector spaces over R and those over C . I shall try to make the distinction when it matters but, if the two cases are treated in the same way, I shall often proceed as above. Although I shall stick with metric spaces as much as possible, there will be points where we shall need the notions of a topological space, a compact topological space and a Hausdorff topological space. I would be happy, if requested, to give a supplementary lecture introducing these notions. (Even where I use them, no great depth of understanding is required.)
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