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. Basic Operator Theory, by I. Gohberg
and S. Goldberg, Birkhauser (Springer). |

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. |