Lecture 4: Hilbert Spaces and Unique Solutions

Introduces continuity in function spaces and the Galerkin method for solving boundary value problems.

Covers the definition and properties of Hilbert spaces, including the Cauchy-Schwarz inequality and norm definition.

Introduces function spaces and Hilbert spaces, discussing inner product spaces and the importance of completeness in Hilbert spaces.

Covers normed spaces, dual spaces, Banach spaces, Hilbert spaces, weak and strong convergence, reflexive spaces, and the Hahn-Banach theorem.

Covers distributions, derivatives, convergence, and continuity criteria in function spaces.