By Elias M. Stein

This can be the fourth and ultimate quantity within the Princeton Lectures in research, a chain of textbooks that target to give, in an built-in demeanour, the center parts of study. starting with the elemental proof of practical research, this quantity seems at Banach areas, Lp areas, and distribution thought, and highlights their roles in harmonic research. The authors then use the Baire type theorem to demonstrate numerous issues, together with the lifestyles of Besicovitch units. the second one 1/2 the ebook introduces readers to different relevant themes in research, corresponding to chance thought and Brownian movement, which culminates within the resolution of Dirichlet's challenge. The concluding chapters discover a number of complicated variables and oscillatory integrals in Fourier research, and illustrate purposes to such diversified parts as nonlinear dispersion equations and the matter of counting lattice issues. during the ebook, the authors specialise in key leads to every one region and tension the natural harmony of the subject.

A finished and authoritative textual content that treats the various major subject matters of recent analysis
A examine uncomplicated sensible research and its functions in harmonic research, chance concept, and several other complicated variables
Key leads to every one region mentioned relating to different components of mathematics
Highlights the natural solidarity of huge parts of study regularly cut up into subfields
Interesting routines and difficulties illustrate ideas
Clear proofs provided

Elias M. Stein is the Albert Baldwin Dod Professor of arithmetic at Princeton collage. Rami Shakarchi obtained his PhD in arithmetic from Princeton collage. they're the coauthors of complicated research, Fourier research, and genuine research (all Princeton).

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Extra info for Functional Analysis:Introduction to Further Topics in Analysis

Sample text

We see that If IL' = 1, and also I = /2(E) fE 2 - 4 The dual space of U when 1

1 (so q < oc), we take fore. I 'I and = I = lIgnhILq this does not exceed M. By lemma it follows that f

Then the dual is equivalent to 12". * There are generalizations of the parallelogram law for L2 (see Exercise 25) that hold for These are the Clarkson inequalities: (a) For 2

G E 17 defines a metric on 17(X) 5. Let X he a measure space Using the argument to prove the completeness of norni, then a converges to f in the show that if the sequence subsequence of f almost everywhere 6. Let (X. F, ii) he a measure space. Show that. (a) The siniple functions are dense in (h) if p(X) < 00, and; simple functions are dense in 17(X) for 1 p < 00

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