## 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
- 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
```