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Real and Convex Analysis

Erhan Çinlar and Robert J. Vanderbei
Publisher: 
Springer
Publication Date: 
2013
Number of Pages: 
161
Format: 
Hardcover
Series: 
Undergraduate Texts in Mathematics
Price: 
49.95
ISBN: 
9781461452560
Category: 
Textbook
[Reviewed by
Allen Stenger
, on
04/18/2014
]

This is a concise introduction to real and abstract analysis, with some convex optimization thrown in. It hits all the high points and famous theorems, but is weak on examples, exercises, and applications.

The first half of the book covers the main points of real functions, although it works primarily in a metric space context and the flavor is more topological than analytic. There is a chapter on differential and integral equations that covers Fredholm and Volterra equations and gives the existence theorems. The approach is ad hoc through contraction mappings rather than in the more general context of functional analysis.

There is an interesting interlude on convex functions and convex sets, which are treated in the context of Euclidean spaces, but the results are fairly general. This is then specialized to the linear programming problem, although only as far as existence and nature of solutions: there is no simplex method and no specific examples.

The book finishes with a more abstract treatment of measure and integral in the context of arbitrary sets and sigma algebras. This is covered in complete generality, and there seems to be only one page on the Lebesgue integral on the real line (and this is to contrast it with the Riemann integral and give the criterion for a function to be Riemann-integrable).

Bottom line: a clean and easy-to-follow introduction, covering all the most important facts, but I would have liked more concrete examples.


Allen Stenger is a math hobbyist and retired software developer. He is webmaster for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.

​​Sets and Functions

Metric Spaces

Functions on Metric Spaces

Differential and Integral Equations

Convexity

Convex Optimization

Measure and Integration

Bibliography

Index

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