ABSTRACT

A modern introduction to the theory of real variables and its applications to all areas of analysis and partial differential equations. The book discusses the foundations of analysis, including the theory of integration, the Lebesque and abstract integrals, the Radon-Nikodym Theorem, the Theory of Banach and Hilbert spaces, and a glimpse of Fourier series. All material is presented in a clear and motivational fashion.

chapter 1|14 pages

Cardinal Numbers

chapter 2|12 pages

Ordinal Numbers

chapter 3|18 pages

The Riemann-Stieltjes Integral

chapter 4|18 pages

Abstract Measures

chapter 5|16 pages

The Lebesgue Measure

chapter 6|26 pages

Measurable Functions

chapter 7|26 pages

Integration

chapter 8|18 pages

More About L1

chapter 9|16 pages

Borel Measures

chapter 10|18 pages

Absolute Continuity

chapter 11|26 pages

Signed Measures

chapter 12|28 pages

L p Spaces

chapter 13|30 pages

Fubini’s Theorem

chapter 14|30 pages

Normed Spaces sind Functionals

chapter 15|20 pages

The Basic Principles

chapter 16|40 pages

Hilbert Spaces

chapter 17|14 pages

Fourier Series

chapter 18|28 pages

Remarks on Problems and Questions