## 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