文件名称：real and complex analysis-Walter rudin
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实分析与复分析
In this book I present an analysis course which I have teach to first year graduate students at the Univereity of Wisconsin since 1962. The course was developed for two reasons. The first was a belief that one could present the basic techniques and theorems of analysis in one year, with enough applications to make the subject interesting, in such a way that students could then specialize in any direction they choose. The second and perhaps even more important one was the desire to do away with the outmoded and misleading idea that analysis consists of two distinct halves, "real variables" and "complex variables.'' Traditionally (with some oversimplification) the first of these deals with Lebesgue integration, with various types of convergence, and with the pathologies exhibited by very discontinuous functions; whereas the second one concerns itself only with those functions that are as smooth rts can be, namely, the holomorphic ones. That these two areas interact most intimately has of course been well known for at least 60 years and is evident to anyone who is acquainted with current research. Nevertheless, the standard curriculum in most American universities still contains a year course in complex variables, followed by a year course in real variables, and usually neither of these courses acknowledges the existence of the subject matter of the other. I have made an effort to demonstrate the interplay among the various parts of analysis, including some of the basic ideas from functional analysis. Here are a few examples. The Riesz representation theorem and the Hahn-Banach theorem allow one to "guess" the Poisson integral formula. They team up in the proof of Runge's theorem, from which the homology version of Cauchy's theorem follows easily. They combine with Blaschke's theorem on the zeros of bounded holomorphic functions to give a proof of the Miintz-Szasz theorem, which concerns approximation on an interval. The fsct that LZ is a Hilbert space is used in the proof of the W o n - N i i y m theorem, which leads to the theorem ,about differentiation of indefinite integrals (incidentally, daerentiation seems to be unduly slighted in most modern texts), which in turn yields the existence of radial limits of bounded harmonic functions. The theorems of Plancherel and Cauchy combined give a theorem of Paley and Wiener which, in turn, is used in the Denjoy-Carleman theorem about infinitely differentiable functions on the real lime. The maximum modulus theorem gives information about linear transformations on Lp-spsces. Since most of the results presented here are quite classical (the novelty lies in the arrangement, and some of the proofs are new), I have not attempted to document the source of every item. References are gathered at the end, in Notes and Comments. They are not always to the original sources, but more often to more recent works where further references can be found. In no case does the absence of a reference imply any claim to originality on my part. The prerequisite for this book is a good course in advanced calcuIus (set-theoretic manipulations, metric spaces, uniform continuity, and uniform convergence). The first seven chapters of my earlier book "Principles of Mathematical Adysis" furnish smcient preparation.