Exercises in Functional Analysis (Texts in the Mathematical Sciences (26))

This book contains almost 450 exercises, all with complete solutions; it provides supplementary examples, counter-examples, and applications for the basic notions usually presented in an introductory course in Functional Analysis. Three comprehensive sections cover the broad topic of functional analysis. A large number of exercises on the weak topologies is included. The understanding of results and notions for a student in mathematics requires solving ex­ ercises. The exercises are also meant to test the reader's understanding of the text material, and to enhance the skill in doing calculations. This book is written with these three things in mind. It is a collection of more than 450 exercises in Functional Analysis, meant to help a student understand much better the basic facts which are usually presented in an introductory course in Functional Analysis. Another goal of this book is to help the reader to understand the richness of ideas and techniques which Functional Analysis offers, by providing various exercises, from different topics, from simple ones to, perhaps, more difficult ones. We also hope that some of the exercises herein can be of some help to the teacher of Functional Analysis as seminar tools, and to anyone who is interested in seeing some applications of Functional Analysis. To what extent we have managed to achieve these goals is for the reader to decide. This Book Of Exercises In Functional Analysis Contains Almost 450 Exercises (all With Complete Solutions), Providing Supplementary Examples, Counter-examples And Applications For The Basic Notions Usually Presented In An Introductory Course In Functional Analysis. It Contains Three Parts. The First One Contains Exercises On The General Properties For Sets In Normed Spaces, Linear Bounded Operators On Normed Spaces, Reflexivity, Compactness In Normed Spaces, And On The Basic Principles In Functional Analysis: The Hahn-banach Theorem, The Uniform Boundedness Principle, The Open Mapping And The Closed Graph Theorems. The Second One Contains Exercises On The General Theory Of Hilbert Spaces, The Riesz Representation Theorem, Orthogonality In Hilbert Spaces, The Projection Theorem And Linear Bounded Operators On Hilbert Spaces. The Third One Deals With Linear Topological Spaces, And Includes A Large Number Of Exercises On The Weak Topologies.