Preface

The purpose of this book is to present some mathematical tools that play key roles in mathematics as well as in applied mathematics, physics, and engineering. The treatment is mathematical in nature, and we do not go into concrete applications; but it is important to stress that all the considered topics are selected because they actually play a role outside pure mathematics. The hope is that the book will be useful for students in many fields of science and engineering, and professionals who want a deeper insight in some of the topics appearing in the scientific literature. A central theme throughout the work is the structure of various vector spaces (most importantly, normed vector spaces and Hilbert spaces) and expansions of elements in these spaces in terms of bases. Particular attention is given to the space of square-integrable functions, L(R). The goal is twofold. Besides the interest in these subjects by themselves, the book will also contribute to a deeper understanding of several themes from calculus and linear algebra, because these themes appear here again and are tied together. For example, we discuss Fourier series in the correct setting of an expansion in a Hilbert space, similar to the one that is obtained via an orthonormal basis in C.