Elements Of Functional Analysis And Operator Theory

This book is designed as a two-semester text. The first semester is devoted to Banach and Hilbert spaces, while the second semester focuses on operator theory.The book aims not only to present the core concepts in a clear and concise manner, but also to enrich the reader's understanding through numerous illustrations and a wide range of exercise problems. Special focus has been laid on important theorems like open map theorem, closed graph theorem, Hahn-Banach theorems, principle of uniform boundedness, etc, which play a crucial role in the study of functional analysis. Moreover, the reader will also find brief discussions on various tricky topics like comparison between two types of adjoint operators — Hilbert space adjoint and Banach space adjoint, etc. Careful attention has been paid on the hypothesis of the results and counterexamples have been provided for their significance.The prerequisites for this book include undergraduate courses in real analysis, linear algebra and basic point set topology (for example, metric spaces). Beyond this, some familiarity with measure theory and Lebesgue integration is desirable, but not essential. Most of the use of measure theory and Lebesgue integration occurs in limited ways.