This course will be an introduction, in the spirit of a user’s guide, to modern techniques in analysis, which are central to the theoretical and numerical treatment of random systems.
This is an 8 hour course held in the first two weeks of the CDT in Random Systems
Prerequisites: Basic Functional Analysis and Lebesgue Integration.
Prof. Gui-Qiang Chen
Learning Outcomes Students will learn basic techniques and results about Lebesgue and Sobolev spaces, distributions and weak derivatives, embedding and trace theorems, and weak convergence.
Revision of relevant definitions and statements from functional analysis: completeness, separability, compactness and duality.
Revision of relevant definitions and statements from Lebesgue integration theory: convergence theorems, completeness, separability and duality.
Weak and weak* convergence in Lebesgue spaces: oscillation and concentration. Equi-integrability and Vitali's Convergence Theorem. Examples. A bounded sequence in the dual of a separable Banach space has a weak* convergent subsequence. Statement of Mazur's Lemma.
Mollifiers and the density of smooth functions
Vitali's covering lemma and maximal inequalities. Lebesgue points and precise representatives.
Distributions and distributional derivatives. Positive distributions are measures and statements of the Riesz representation theorem.
Sobolev spaces: mollifications and weak derivatives, separability and completeness. Poincaré and Sobolevinequalities. Embedding theorems and Rellich-Kondrachov-Sobolev theorems on compactness (sketches of proofs only).
Traces of functions with weak derivatives.