Required Functional Analysis

Required Functional Analysis#

In the class lectures, we will cover the basics of functional analysis. I am not going to repeat the material here as it is not my intention to write a textbook on functional analysis. There are excellent books out there. The material I am going to cover in class is based on Introductory Functional Analysis with Applications by Erwin Kreyszig. Specifically, I will cover the following topics in class:

  • 1.1. Metric Space

  • 1.2. Further Examples of Metric Spaces

  • 1.3. Open Set, Closed Set, Neighborhood

  • 1.4. Convergence, Cauchy Sequence, Completeness

  • 2.1. Vector Space

  • 2.2. Normed Space. Banach Space

  • 3.1. Inner Product Space. Hiblert Space

  • 3.2. Further Properties of Inner Product Spaces

  • 3.3. Orthogonal Complements and Direct Sums

  • 3.4. Orthonormal Sets and Sequences

  • 3.5. Series Related to Orthonormal Sequences

  • 3.6. Total Orthonormal SEts and Sequences

  • 3.7. Legendre, Hermite and Laguerre Polynomials

This is the minimum required to understand polynomial chaos. It’s about 150 pages and we will cover it in four lectures. The theory here is extremely useful in many areas of scientific machine learning. Apart from polynomial chaos, it is needed in proper orthogonal decomposition, Koopman theory, operator learning, and many other areas. The majority of scientific machine learning papers make frequent references to concepts from functional analysis.