Description - Real Analysis by Mark Bridger
A unique approach to analysis that lets you apply mathematics
across a range of subjects
This innovative text sets forth a thoroughly rigorous modern
account of the theoretical underpinnings of calculus: continuity,
differentiability, and convergence. Using a constructive approach,
every proof of every result is direct and ultimately
computationally verifiable. In particular, existence is never
established by showing that the assumption of non-existence leads
to a contradiction. The ultimate consequence of this method is that
it makes sense?not just to math majors but also to students
from all branches of the sciences.
The text begins with a construction of the real numbers
beginning with the rationals, using interval arithmetic. This
introduces readers to the reasoning and proof-writing skills
necessary for doing and communicating mathematics, and it sets the
foundation for the rest of the text, which includes:
Early use of the Completeness Theorem to prove a helpful
Inverse Function Theorem
Sequences, limits and series, and the careful derivation of
formulas and estimates for important functions
Emphasis on uniform continuity and its consequences, such as
boundedness and the extension of uniformly continuous functions
from dense subsets
Construction of the Riemann integral for functions uniformly
continuous on an interval, and its extension to improper
integrals
Differentiation, emphasizing the derivative as a function
rather than a pointwise limit
Properties of sequences and series of continuous and
differentiable functions
Fourier series and an introduction to more advanced ideas in
functional analysis
Examples throughout the text demonstrate the application of new
concepts. Readers can test their own skills with problems and
projects ranging in difficulty from basic to challenging.
This book is designed mainly for an undergraduate course, and
the author understands that many readers will not go on to more
advanced pure mathematics. He therefore emphasizes an approach to
mathematical analysis that can be applied across a range of
subjects in engineering and the sciences.
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