Function Space Methods for Systems Theory, EECS/IOE 600, Winter 2021

This course will give you fundamental mathematical tools that are used in systems theory-- signal processing, control, and optimization. Be prepared for a good time practicing abstract thinking and learning how to construct a beautiful proof!

You can find the full syllabus here or on the canvas site. If you ever have questions about the course material or syllabus, ask them on Piazza.

Instructor: Professor Laura Balzano,
Course time: Tuesday/Thursday 10:30am-12pm, January 19 - April 21
Course location: Remote via Zoom. All lectures will be recorded.
Office hours: TBD
Textbook: Optimization by Vector Space Methods by Luenberger.
Tentative syllabus: We will cover the following topics in roughly this order.

Vector spaces
 - definition
 - subspaces, linear combinations, span
 - linear independence and basis
 - subspaces, cones, convexity
 - functions and mappings
 - vector spaces of mappings

Metric spaces
 - topology
 - sequences and convergence in metric spaces
 - fixed points and contraction mappings
 - continuous functions

Normed Vector spaces 
 - Banach spaces
 - lp and Lp spaces
 - projections
 - applications (signal processing, control, optimization)

Inner Product spaces
 - Hilbert spaces
 - orthogonal complement, orthogonality principle
 - function approximation
 - projections
 - applications (signal processing, control, optimization)

Linear functions
 - Linear functionals
 - Dual space
 - Riesz Representation theorem
 - Hahn-Banach theorem
 - Linear operators
 - adjoint operators
 - solving systems of linear equations

Optimization of functionals
 - conjugate functional
 - Minkowski functional
 - Fenchel duality

With time:

Spectral theory
 - The Spectral theorem
 - eigenvalues and eigenvectors of integral operators

Optimization using lagrange multipliers
 - sufficient conditions for arbitrary functions
 - convex functions
 - necessary conditions for convex functions
 - Gateaux derivative
 - Kuhn-Tucker sufficiency
 - Frechet derivative

More results in optimization
 - Farkas lemma
 - Karush-Kuhn-Tucker theorem