The purpose of this book is to describe an approach to maximum and minimum principles which is both straightforward and unified. A particular aim is to identify and illustrate the structure of the theory, and to show how that structure leads to general formulae for the construction of upper and lower bounds associated with those principles. Such bounds are important in applications.

I have found that fresh and fruitful insights have arisen repeatedly during the working out of the ideas developed in this approach. It is offered to the reader in the hope that, as he absorbs the viewpoint, he will benefit from similar experiences.

The treatment is designed to be accessible in the first three chapters to final year undergraduates in mathematics and science, and throughout to contain material which will interest postgraduates and research workers in those subjects. Some elementary prior knowledge of the calculus of variations will be helpful, but otherwise the book is self-contained. Anyone knowing more than this minimum should be able to read Chapter 3 first, with occasional references back to Chapter 1.1 have given a central role to a pair of inner product spaces and, by emphasizing this simple idea, I have been able to avoid the need for any sophisticated functional analysis. The reader who does possess more technical knowledge may use it to add to what is here; he will recognise topics which I have omitted. The reader without such extra knowledge will be at no disadvantage in reading the book.