A Concise Introduction to Functional Analysis is designed to serve a one-semester introductory graduate (or advanced undergraduate) course in functional analysis.
The text is pragmatically structured so that each unit corresponds to one class, with the hope of being helpful for both students and teachers. It is expected that this text will provide students with a strong general understanding of the subject, and that they should feel well equipped to take on the more advanced texts and courses covering topics not treated here.
Features
· Numerous examples and counterexamples to illustrate such abstract concepts
· Over 430 exercises, with partial solutions included in the book itself
· Minimal pre-requisites beyond linear algebra and general topology.
A Concise Introduction to Functional Analysis is designed to serve a one-semester introductory graduate (or advanced undergraduate) course in functional analysis.
Preface Selected Notation
Chapter 1 Normed Spaces
Chapter 2 Compactness
and Completion
Chapter 3 Separable Spaces and Linear Operators
Chapter 4
Bounded Operators and Dual Spaces
Chapter 5 Banach Fixed Point
Chapter 6
Baire Theorem
Chapter 7 Uniform Boundedness Principle
Chapter 8 Open Mapping
Theorem
Chapter 9 Closed Graph Theorem
Chapter 10 Hahn-Banach Theorem
Chapter
11 Proof of Hahn-Banach
Chapter 12 Applications of Hahn-Banach Theorem
Chapter 13 Adjoint Operators in N
Chapter 14 Weak Convergence
Chapter 15 Weak
Topologies
Chapter 16 Reflexive Spaces and Compactness
Chapter 17 Hilbert
Spaces
Chapter 18 Orthogonal Projection
Chapter 19 Riesz Representation in H
Chapter 20 Self-Adjoint Operators
Chapter 21 Orthonormal Bases
Chapter 22
Fourier Series
Chapter 23 Operations on Banach Spaces
Chapter 24 Compact
Operators
Chapter 25 Compact Operators on H
Chapter 26 Hilbert-Schmidt
Operators
Chapter 27 The Spectrum
Chapter 28 Spectral Classification
Chapter
29 Spectra of Self-Adjoint Operators
Chapter 30 Spectra of Compact Operators
Solutions to Selected Exercises Bibliography Index
César R. de Oliveira earned his Ph.D. in Physics from the University of São Paulo in 1987. He has been a visiting professor at the Università degli Studi di Milano (19911992) and the University of British Columbia (20082009). He is currently a Full Professor in the Federal University of São Carlos.
His research lies in the field of Mathematical Physics, with publications in both mathematics and physics journals. He has supervised twelve Ph.D. students, and his main areas of interest include Schrödinger and Dirac operators, the AharonovBohm effect, mathematical models of graphene, quantum (in)stability, and dynamical localization.
He also enjoys writing textbooks and has authored four of them. In 2000, he published an introductory mechanics book (in Portuguese, with animations and interactive content on CD-ROM) through his University. In 2010, he released a graduate-level book on functional analysis (in Portuguese) with IMPA (Rio de Janeiro). In 2009, the book Spectral Theory and Quantum Dynamics was published by Birkhäuser (Switzerland). Most recently, in 2023, he co-authored Spectral Measures and Dynamics: Typical Behaviors with M. Aloisio and S. Carvalho.
