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MATH622: Introduction to Functional Analysis
Revised: November 2006 (Erin McNelis)
Course Description & Topics
This first course in functional analysis will focus on:
- spaces and their operators
- metric spaces and metrics (Chapter 1)
- vector, normed and Banach spaces, and norms and linear operators (Chapter 2)
- inner product and Hilbert spaces, and inner products (Chapter 3)
- fundamental theorems (Chapter 4)
Course Objectives
To provide students with a strong foundation in functional analysis, focusing on spaces and operators, fundamental theorems; to strengthen students understanding of this theory through applications of functional analysis; to develop students skills and confidence in mathematical analysis and proof techniques. To build an understanding of mathematical analysis in Rn through the use of mathematical proof.
Text
Kreyszig. Introductory Functional Analysis with Applications (Wiley Classics Library), John Wiley & Sons, Inc, 1978.
Grading Procedure
Grading procedures and factors influencing course grade are left to the discretion of individual instructors, subject to general university policy. Commonly, the final grade will be based upon homework/labwork, in-class tests, a class journal, and a final project.
Attendance Policy
Attendance policy is left to the discretion of individual instructors, subject to general university policy.
Course Outline
Metric Spaces (Chapter 1) [9 days]
- Metric Spaces and Examples of Metric Spaces
- Open/Closed Sets and Neighborhoods
- Convergence, Cauchy Sequences, and Completeness
- Completion of Metric Spaces
Normed Spaces (Chapter 2) [15 days]
- Vector Spaces
- Normed Spaces and Banach Spaces
- Finite Dimensional Normed Spaces
- Compactness
- Linear Operators
- Bounded and Continuous Linear Operators
- Linear Functionals
- Dual Spaces
Inner Product Spaces (Chapter 3) [16 days]
- Inner Product and Hilbert Spaces
- Orthogonal Complements and Direct Sums
- Orthonormal Sets and Sequences
- Total Orthonormal Sets and Sequences
- Representation of Functionals on Hilbert Spaces
- Hilbert-Adjoint Operators
- Self-Adjoint, Unitary and Normal Operators
Fundamental Theorems and Applications (A Portion of Chapter 4) [5 days]
- Zorn's Lemma
- Hahn-Banach Theorem
- Application to Bounded Linear Functionals on C[a, b]