Bolzano-Weierstrass Theorem: Sequential Compactness in Hilbert Spaces

This lecture covers the Bolzano-Weierstrass theorem, stating that the closed unit disk in a Hilbert space is sequentially compact. The proof involves constructing subsequences that converge weakly, leading to the existence of a unique limit point. By iteratively building subsequences, a linear bounded functional is defined, extending to the entire space. The lecture concludes by discussing the limitations of the theorem and providing exercises to illustrate its application.