Lecture
This lecture covers the extension of bounded linear transformations defined on dense subspaces, with applications to the extension of the Fourier transform to L^2 spaces. It introduces the free propagator P_f on L^2, which is unitary, bicontinuous, and satisfies the group composition law. The lecture also defines bicontinuous one-parameter groups and shows how P_f provides a distributional solution to the free Schrödinger equation. Additionally, it discusses the definition of Sobolev spaces and their properties, such as strong continuity and self-duality.