id: 06532701
dt: b
an: 2016c.00854
au: Simon, Barry
ti: Real analysis. A comprehensive course in analysis, part 1.
so: Providence, RI: American Mathematical Society (AMS) (ISBN
978-1-4704-1099-5/hbk). xx, 789~p. (2015).
py: 2015
pu: Providence, RI: American Mathematical Society (AMS)
la: EN
cc: I55
ut: measure theory; distribution theory; Banach space; Schwartz space; fixed
point; Haar measure
ci: Zbl 1332.00004; Zbl 1332.00005; Zbl 1334.00002; Zbl 1334.00003; Zbl
1331.00001
li: doi:10.1090/simon/001
ab: The volume under review presents a clear, thorough treatment of the
theorems and concepts of real analysis. The author is concerned with
the fundamentals and touchstone results of the real analysis in full
rigor, but in a style that requires little prior familiarity with
proofs or mathematical language. The book develops those concepts and
tools in real analysis that are vital to every mathematician, whether
pure or applied, aspiring or established. These works present a
comprehensive treatment with a global view of the subject, emphasizing
the connections between real analysis and other branches of
mathematics. From a first point of view, the volume presents the
infinitesimal calculus of the twentieth century with the ultimate
integral calculus (measure theory) and the ultimate differential
calculus (distribution theory). From another point of view, it shows
the triumph of abstract spaces such as topological spaces, Banach and
Hilbert spaces, measure spaces, Riesz spaces, Polish spaces, locally
convex spaces, Fréchet spaces, Schwartz space, and Lebesgue
$L^p$-spaces. The text includes many elegant proofs and an excellent
choice of topics, including the Fourier series and transform, dual
spaces, the Baire category, fixed point theorems, probability ideas,
and Hausdorff dimension. The book includes plenty of relevant
applications, for instance the constructions of nowhere differentiable
functions, Brownian motion, space-filling curves, solutions of the
moment problem, Haar measure, and equilibrium measures in potential
theory. Numerous examples and exercises are integrated into the main
text as well as the end of each chapter to reinforce the methodology.
With clear proofs, detailed examples, and numerous exercises, this book
gives a thorough treatment of the subject. It provides a logical
development of material that will prepare readers for more advanced
analysis-based studies. This volume prepares the reader for further
exploration of measure theory, functional analysis, harmonic analysis,
and beyond. The clarity and breadth of {\it Real analysis. A
comprehensive course in analysis, part 1} make this volume a welcome
addition to the personal library of every mathematician.
rv: Teodora-Liliana Rădulescu (Craiova)