A First Course in Functional Analysis 1st Edition by Orr Moshe Shalit, ISBN-13: 978-1498771610
[PDF eBook eTextbook]
- Publisher: Chapman and Hall/CRC; 1st edition (March 7, 2017)
- Language: English
- 240 pages
- ISBN-10: 9781498771610
- ISBN-13: 978-1498771610
Written as a textbook, A First Course in Functional Analysis is an introduction to basic functional analysis and operator theory, with an emphasis on Hilbert space methods. The aim of this book is to introduce the basic notions of functional analysis and operator theory without requiring the student to have taken a course in measure theory as a prerequisite. It is written and structured the way a course would be designed, with an emphasis on clarity and logical development alongside real applications in analysis. The background required for a student taking this course is minimal; basic linear algebra, calculus up to Riemann integration, and some acquaintance with topological and metric spaces.
Table of Contents:
Cover
Half Title
Title Page
Dedication
Table of Contents
Preface
1 Introduction and the Stone-Weierstrass theorem
1.1 Background and motivation
1.2 The Weierstrass approximation theorem
1.3 The Stone-Weierstrass theorem
1.4 The Stone-Weierstrass theorem over the complex numbers
1.5 Concluding remarks
1.6 Additional exercises
2 Hilbert spaces
2.1 Background and motivation
2.2 The basic definitions
2.3 Completion
2.4 The space of Lebesgue square integrable functions
2.5 Additional exercises
3 Orthogonality, projections, and bases
3.1 Orthogonality
3.2 Orthogonal projection and orthogonal decomposition
3.3 Orthonormal bases
3.4 Dimension and isomorphism
3.5 The Gram-Schmidt process
3.6 Additional exercises
4 Fourier series
4.1 Fourier series in L2
4.2 Pointwise convergence of Fourier series (Dirichlet’s theorem)
4.3 Fejér’s theorem
4.4 *Proof of Dirichlet’s theorem
4.5 Additional exercises
5 Bounded linear operators on Hilbert space
5.1 Bounded operators
5.2 Linear functionals and the Riesz representation theorem
5.3 *The Dirichlet problem for the Laplace operator
5.4 The adjoint of a bounded operator
5.5 Special classes of operators
5.6 Matrix representation of operators
5.7 Additional exercises
6 Hilbert function spaces
6.1 Reproducing kernel Hilbert spaces
6.2 *The Bergman space
6.3 *Additional topics in Hilbert function space theory
6.4 Additional exercises
7 Banach spaces
7.1 Banach spaces
7.2 Bounded operators
7.3 The dual space
7.4 *Topological vector spaces
7.5 Additional exercises
8 The algebra of bounded operators on a Banach space
8.1 The algebra of bounded operators
8.2 An application to ergodic theory
8.3 Invertible operators and inverses
8.4 Isomorphisms
8.5 Additional exercises
9 Compact operators
9.1 Compact operators
9.2 The spectrum of a compact operator
9.3 Additional exercises
10 Compact operators on Hilbert space
10.1 Finite rank operators on Hilbert space
10.2 The spectral theorem for compact self-adjoint operators
10.3 The spectral theorem for compact normal operators
10.4 The functional calculus for compact normal operators
10.5 Additional exercises
11 Applications of the theory of compact operators
11.1 Integral equations
11.2 *Functional equations
11.3 Additional exercises
12 The Fourier transform
12.1 The spaces Lp(ℝ), p ∈ [1, ∞)
12.2 The Fourier transform on LX(ℝ)
12.3 The Fourier transform on L2(ℝ)
12.4 *Shannon’s sampling theorem
12.5 *The multivariate Fourier transforms
12.6 Additional exercises
13 *The Hahn-Banach theorems
13.1 The Hahn-Banach theorems
13.2 The dual space, the double dual, and duality
13.3 Quotient spaces
13.4 Additional excercises
Appendix A Metric and topological spaces
A.1 Metric spaces
A.2 Completeness
A.3 Topological spaces
A.4 The Arzelà-Ascoli theorem
Bibliography
Index
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