This book offers an introduction to some basic aspects of modern analysis. It is designed for students who are majoring in some area of mathematics but who do not necessarily intend to continue their studies at a graduate level.

The choice of material and the method of presentation are both aimed at as wide a readership as possible. Future teachers of high school mathematics should be given an introduction to the mathematical future as much as they must be given some knowledge of the mathematical past; students of mathematical engineering, biology or finance may need to read current literature without desiring to contribute to it. These are perhaps the extremes in the type of student to whom this book is directed. At the same time, students who do need to go on to courses in measure theory and functional analysis will find this book an easy introduction to the initial concepts in those areas.

Syllabus requirements would rarely allow more than one semester to be available for work of this nature in an undergraduate course and this imposes restrictions on topics and the depth of their presentation. In line with the above thoughts, I have tried throughout to merge the nominal divisions of pure and applied mathematics, leaving enough for students of either inclination to have a feeling for what further developments might look like. After a somewhat objective choice of topics, the guiding rule in the end was to carry those topics just far enough that their applications might be appreciated. Applications have been included from such fields as differential and integral equations, systems of linear algebraic equations, approximation theory, numerical analysis and quantum mechanics.