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Unformatted text preview: Linear Analysis T. W. K¨orner January 8, 2008 Small print The syllabus for the course is defined by the Faculty Board Schedules (which are minimal for lecturing and maximal for examining). Several of the results are called Exercises. I will do some as part of the lectures but others will be left to the reader. In general these are simple verifications or recall results from earlier courses. If you find that you cannot do one of these, consult your supervisor. I have sketched solutions for supervisors to the exercises in the five example sheets at the end. These should be available from the departmental secretaries or in tex, ps, pdf and dvi format by email. 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. 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/ My email address is [email protected] . Contents 1 Some inequalities 2 2 In finite dimension, norms are equivalent 5 3 Banach spaces 7 4 Continuous linear functions 9 5 Second duals 11 6 Baire category 13 7 Continuous functions 17 8 The Stone–Weierstrass theorem 19 1 9 Ascoli–Arzel` a 21 10 Inner product spaces 23 11 Hilbert space 26 12 The dual of Hilbert space 29 13 The spectrum 30 14 Selfadjoint compact operators on Hilbert space 31 15 Using the spectral theorem 34 16 Where next? 37 17 Books 40 18 First example sheet 40 19 Second example sheet 45 20 Third example sheet 50 21 Fourth example sheet 54 22 Supplementary exercises 58 1 Some inequalities Inequalities lie at the heart of analysis. In this section we prove some in equalities which lie at the heart of this course. We start with some observations from Part 1A. Exercise 1.1. Suppose f : (0 , ∞ ) → R is twice differentiable with f ′′ ( x ) < for all x ∈ R . Then if < t < s and < λ < 1 , it follows that λf ( t ) + (1 − λ ) f ( s ) < f ( λt + (1 − λ ) s ) . In other words, if f ′′ ( x ) < 0, then f is strictly concave. Applying the result with f = log, we obtain the following result. 2 Lemma 1.2. Suppose p, q are real and positive with 1 p + 1 q = 1 . Then, if a, b > , we have a 1 /p b 1 /q ≤ a p + b q with equality if and only if a = b . Our result remains true if a, b ≥ . We can now obtain our first version of the H¨older inequality 1 . (Here and elsewhere we will write F to mean either R or C . This reflects the fact that many theorems of Linear Analysis apply both to real and complex vector spaces. However, just as in other branches of analysis and algebra, there are important theorems which apply only to complex or only to real spaces.) Theorem 1.3. Suppose p, q are real and positive with 1 p + 1 q = 1 ....
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 Fall '08
 Groah
 Math, Space, Metric space, Hilbert space, Compact space, Banach

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