Orthonormal Sequences and Hilbert Spaces

This lecture covers the concept of orthonormal sequences in Hilbert spaces, explaining how they can be used to decompose elements into orthogonal components. It also delves into Bessel's inequality, Cauchy-Schwarz inequality, and the importance of separable Hilbert spaces. The instructor discusses the relationship between orthonormal bases and algebraic bases, the polarization identity, and the characterization of orthonormal bases. The lecture concludes with a detailed proof involving orthogonal complements and unique decompositions in Hilbert spaces.