This book represents material suitable for a two semester graduate
course in analysis. Based on courses given by the author since
2007, it is targeted towards graduate students preparing for a
research career in mathematics. After reviewing what a typical
student would have learned in single variable and multivariable
calculus courses, the first chapter presents familiar material from
a mature mathematical perspective. The second chapter surveys
measure theory and highlights the essential theorems that have now
become ubiquitous in mathematics. The third chapter on Fourier
transforms derives the inversion theorem and gives diverse
applications ranging from probability theory to mathematical
physics. The fourth chapter reviews complex analysis giving a
synthetic, rigorous development of the calculus of residues and its
applications to a wide array of problems. The final chapter is an
introduction to algebraic topology and is included to show the
symbiosis between algebra and analysis.
General
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