#### Prerequisites & Bulletin Description

#### Course Objectives

The aim of Analysis II is to introduce students to the theory and application of basic functional analysis. The course begins with a survey of the theory of complete metric spaces then focuses on Banach and Hilbert spaces, their operators and dual spaces. With classical function spaces as motivating examples, students will learn the concepts, results, and applications of functional analysis.

Students who successfully complete this course will be able to apply results such as the Hahn-Banach theorem, the closed graph theorem, and the Uniform Boundedness Principle in specific contexts such as Fourier analysis, differential and integral equations. They will devise, organize, and present their solutions in correct mathematical English.

#### Evaluation of Students

Students will be graded on their ability to devise, organize and present in correct mathematical English rigorous solutions to assignments and problems on exams. While instructors may design their own methods of evaluating student performance, these methods must include in-class examinations, assignments and a final examination.

#### Course Outline

Topics |
Number of Weeks |

Metric spaces, Baire Category Theorem | 3 |

Banach and Hilbert spaces, classical examples | 3 |

Bounded operators and dual spaces | 3 |

Hahn-Banach, closed graph, and open mapping theorems | 3 |

Applications | 2 |

#### Textbooks & Software

*Introductory Functional Analysis with Applications *, by Kreyszig

Updated by: Eric Hayashi

Date: March, 2011