MATH 6220 - Applied Functional Analysis

Yury Kudryashov, spring 2017.

Functional analysis deals with linear algebra and analysis in infinite-dimensional vector spaces, e.g., spaces of functions. It turns out that most of the finite dimensional intuition does not work in this case. For example, not all linear operators are continuous.

Applications of functional analysis include (but are not limited to):

• Optimization problems, e.g.
  – how to find the shortest path connecting two points on a surface?
  – how a beam of light will go in a heterogeneous medium?
• PDEs.
• Mathematical formulation of quantum mechanics.

We shall concentrate on the applications, and study theoretical aspects as required. The theoretical aspects will include:

• Continuous linear operators.
• Hilbert spaces, and the spectrum of a self-adjoint operator.
• Hahn-Banach Theorem and Optimization.
• Variational principles.