Spring 2019     Functional Analysis MTH 629

 Instructor Orlando Merino, merino@math.uri.edu, 874-4442, 200F Lippitt Hall Meets MWF 2-2:50, Lippitt 201 Text Introductory Functional Analysis with Applications, by E. Kreyszig, 1989, J. Wiley. ISBN: 978-0471-50459-7 Prerequisites Linear Algebra MTH513 or equivalent, and familiarity with mathematical proofs. Topics Hilbert and Banach spaces, and continuous linear functions or operators between such spaces. Evaluation Final Exam (25%), Assignments (75%). About the Course The course is an introduction to functional analysis with emphasis on applications. The course is designed for students in mathematics, science, engineering and other fields. We will cover Chapters 1 through 3 from the text, plus some supplementary material. About Functional Analysis Functional Analysis is the study of infinite dimensional vector spaces and functions on these spaces. Functional Analysis provides tools and a foundation for the study of partial differential equations, quantum mechanics, Fourier and wavelet analysis (or harmonic analysis), numerical analysis, approximation theory, and many other fields. Other refs. Introduction to Hilbert Space, by N. Young. Cambridge University Press, 1988. ISBN 0 521 33071 8 Basic Operator Theory, by I. Gohberg and S. Goldberg, Birkhauser (Springer). ISBN: 0817642625, or Basic Classes of Linear Operators By Gohberg Goldberg and Kashooek ISBN-10: 3764369302 | ISBN-13: 978-3764369309 Topics Chapter 1. Metric Spaces Concepts Chapter 2. Normed Spaces and Banach spaces. 2.1 Vector spaces, finite and infinite dimensional, Hamel basis, subspace. 2.2 Normed Vector Spaces, Banach Spaces,  2.3 subspaces, convergent sequences and series, absolute convergence, Schauder basis, separability, completion 2.4 Finite dimensional normed spaces and subspaces. Equivalent norms. Compactness. Continuous maps.  2.6 Linear Operators. Examples. Range and Null Space. Inverse operator.  2.7 Bounded linear operators.  Norm of a bounded linear operator. 2.8 Linear Functionals.  Algebraic Dual space.  Embedding of a space in the second dual. 2.9 Linear operators on finite dimensional spaces. 2.10 Normed spaces of operators. Dual space. Chapter 3. Inner product spaces and Hilbert spaces. 3.1 Inner product space. Hilbert Space. 3.2 Further properties. 3.3 Orthogonal complements and direct sums. 3.4 Orthonormal sets and sequences. 3.5 Series and Orthonormal sequences and sets. 3.6 Total orthonormal sets and sequences. 3.7 Legendre, Hermite and Laguerre Polynomials. 3.8 Representation of Functionals on Hilbert Spaces. 3.9 adjoint operator. 3.10 Self-adjoint, unitary and normal operators. Chapter 4. Additional topics. If there is time, we may discuss one or more of the following: * The spectral theorem of compact self-adjoint operators in Hilbert space. * The Banach contraction principle * Approximation in Hilbert spaces.