This book provides an introduction both to real analysis and to a range of important
applications that depend on this material. Three-fifths of the book is a series of
essentially independent chapters covering topics from Fourier series and polynomial
approximation to discrete dynamical systems and convex optimization. Studying
these applications can, we believe, both improve understanding of real analysis and
prepare for more intensive work in each topic. There is enough material to allow a
choice of applications and to support courses at a variety of levels.