Spectral Theory: Regularity and Embedding

This lecture covers the spectral theory of operators, focusing on regularity properties and embedding theorems. Topics include the definition of Sobolev spaces, compactly supported functions, and the formulation of embedding theorems. The instructor discusses the decay properties of functions, the concept of generalized classes of Sobolev spaces, and proofs of key lemmas. The lecture concludes with applications of the embedding theorems and connections to functional analysis.