Introduction to Functional Analysis
Authors
Description
The book has a very objective proposal, as it seeks to convey an overview of the basic lines of Functional Analysis, preparing students to consult more comprehensive texts. The idea is that each unit of the book makes it more practical for students to read each topic for the first time.
Some of the subjects covered follow the guidelines of the Postgraduate Mathematics program and the author’s personal taste. It is assumed that readers are familiar with Linear Algebra, General Topology and basic results of Functions of One Complex Variable and Measurement and Integration, since these are subjects dealt with in specific disciplines.
The book offers an extended list of exercises and, at the end, the solution of some proposed exercises, the conclusions of which have been used at some point in the text.
Target audience
Higher education
Name: Introduction to Functional Analysis
Author(s): e César R. de Oliveira
Pages: 257
Publication: IMPA, 2018
ISBN: 978-85-244-0453-5
Edition: 1
1 Normed Spaces
2 Compactness and Completeness
2.1 Compactness and Dimension
2.2 Completeness of Normed Spaces
3 Separable Spaces
3.1 Separable Spaces
3.2 Linear Operators
4 Bounded Operators and Dual Space
5 Banach Fixed Point
6 Baire’s theorem
7 Principle of Uniform Limitation
8 Open Application Theorem
9 Closed Graph Theorem
10 Hahn-Banach Theorem
10.1 Max Zorn’s Lemma
10.2 Hahn-Banach
11 Hahn-Banach Demonstration
12 Hahn-Banach applications
13 Adjunct operators in N
14 Weak convergence
15 Weak Topologies
15.1 Weak Topologies
15.2 Alaoglu’s Theorem
16 Reflective Spaces and Compactness
17 Hilbert spaces
17.1 Internal product
17.2 Orthogonality
18 Orthogonal Projection
18.1 Parallelogram Law
18.2 Orthogonal Projection
19 Riesz representation in H
19.1 Riesz representation
19.2 Hilbert and Lax-Milgram adjoint
20 Self-Assembled Operators
21 Ortonormal Bases
22 Fourier Series
22.1 Fourier Series
22.2 Integration in Hilbert Spaces
23 Operators in Banach Spaces
23.1 Direct Sum
23.2 Quotient Space
24 Compact Operators
25 Compact operators in H
26 Hilbert-Schmidt operators
27 Spectrum
28 Spectral Classification
29 Self-Assembly Spectrum
30 Spectrum of Compact Operators
30.1 Compact Operators
30.2 Normal Operators
Solutions to Selected Exercises
Bibliography
Index