Elementary Functional Analysis

topics. However, only a modest preliminary knowledge is needed. In the first chapter, where we introduce an important topological concept, the so-called topological degree for continuous maps from subsets ofRn into Rn, you need not know anything about functional analysis. Starting with Chapter 2, where infinite dimensions first appear, one should be familiar with the essential step of consider­ ing a sequence or a function of some sort as a point in the corresponding vector space of all such sequences or functions, whenever this abstraction is worthwhile.

This concise text provides a gentle introduction to functional analysis. Chapters cover essential topics such as special spaces, normed spaces, linear functionals, and Hilbert spaces. Numerous examples and counterexamples aid in the understanding of key concepts, while exercises at the end of each chapter provide ample opportunities for practice with the material. Proofs of theorems such as the Uniform Bounded Theory, the Open Mapping Theorem, and the Closed Graph Theorem are worked through step-by-step, providing an accessible avenue to understanding these important results. The prerequisites for this book are linear algebra and elementary real analysis, with two introductory chapters providing an overview of material necessary for the subsequent text.