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Methods of Applied Mathematics I
Theory of Bounded Linear Operators
Volume: 25 lectures @ 90 minutes
Syllabus: Download Course Description (PDF)
Textbook: Material from several sources will be used, including:
Jänich, K., Linear Algebra, Springer, Link to publisher's description
Kreyszig, E., Introductory Functional Analysis with Applications, Wiley,
Link to publisher's descriptionStakgold, I., Green's Functions and Boundary Value Problems, Wiley,
Link to publisher's description
Prerequisites: Graduate standing or permission of instructor
Background and Goals: This two-part introduction to advanced mathematical methods aims to provide mathematical foundations to graduate students majoring in both ME and ECE. The topics covered are closely aligned with those of parallel graduate engineering courses.
The present course focuses on the theory of bounded linear operators on Hilbert spaces, in particular matrix calculus and the spectral theory of compact linear operators.
Key Words: Metric spaces, open sets, dense sets and separable metric spaces, sequences in metric spaces and completeness, vector spaces, Banach spaces, Hilbert spaces, Fourier series, Legendre polynomials, multipole expansion, matrices, eigenvalue problem, basic properties of bounded linear operators on Hilbert spaces, the spectrum, Rayleigh-Ritz method, spectral theorem for compact operators, Sturm-Liouville boundary value problems, singular value decompoition for for comapct operators
Detailed Content:
This course gives an introduction to the theory of bounded linear maps on finite- and infinite-dimensional spaces.
In the first part, notions of linear algebra are reviewed and extended to infinite-dimensional vector spaces. This includes concepts such as scalar products, norms and (Schauder-) bases. As an application, Legendre polynomials, introduced as an orthonormalization of the monomials on the interval $[-1,1]$ are introduced, and their role in multipole expansions is explored. Next, Hilbert spaces are introduced, leading to spaces of square-integrable functions and Fourier series. A look back and comaprison of the obtained results with the finite-dimensional cases of linear algebra concludes this part.
The second part focuses on bounded linear maps on (infinite-dimensional) spaces, introducing the matrix elements of such operators and using these to define Hilbert-Schmidt operators for square-summable sequences and square-integrable functions. The notions of inverses and adjoints of bounded linear operators are discussed and the spectrum of such operators is introduced. Compact operators are introduced and, motivated by a question from the theory of partial differential equations, the spectral theorem for compact operators is established.
The last part is dedicated to applications of the spectral theory, including the Rayleigh-Ritz method (applied specifically to Sturm-Liouville eigenvalue problems) and the polar and singular value decompositions of compact operators, which of course includes these decompositions for matrices.
Overview:
The following textbooks are referenced below:
[K] Kreyszig, E., Introductory functional analysis
[S] Stakgold, I., Green's functions and boundary value problems
| Lecture | Lecture Subject |
|---|---|
| 1 | Introduction |
| 2 | Normed Vector Spaces |
| 3 | Bases and Inner Product Spaces |
| 4 | Bases and Inner Product Spaces |
| 5 | Legendre Polynomials and Applications |
| 6 | Legendre Polynomials and Applications |
| 7 | Hilbert Spaces |
| 8 | The Space of Square-Integrable Functions |
| 9 | Fourier Series |
| 10 | Finite-Dimensional Vector Spaces |
| 11 | First Midterm Exam |
| 12 | Linear Functionals and Operators |
| 13 | Matrix Elements and Hilbert-Schmidt Operators |
| 14 | Inverse and Adjoint of Bounded Linear Operators |
| 15 | The Spectrum |
| 16 | The Spectrum |
| 17 | Compact Operators |
| 18 | Spectral Theorem for Compact Operators |
| 19 | Spectral Theorem for Compact Operators |
| 20 | Second Midterm Exam |
| 21 | Sturm-Liouville Boundary Value Problems |
| 22 | The Rayleigh-Ritz Method |
| 23 | Positive Operators and the Polar Decomposition |
| 24 | The Singular Value Decomposition for Compact Operators and Matrices |
| 25 | Final Exam |
Lecture Slides (FDF):
Assignments (PDF):