Introduction to Hilbert Spaces: An Adventure In Infinite Dimensions

About this course:

A Hilbert space is a vector space that is endowed with an inner product for which the corresponding metric is complete (i.e., every Cauchy sequence converges).  Examples include finite-dimensional Euclidean spaces; the space l2 of all infinite sequences (a1, a2, a3, …) of complex numbers, the sum of whose squared moduli converges; and the space L2 of all square-summable functions on an interval. This introductory, yet rigorous, treatment focuses initially on the structure (orthogonality, orthonormal bases, linear operators, Bessel’s inequality, etc.) of general Hilbert spaces, with the latter part of the course devoted to interpreting these constructs in the context of Legendre polynomials, Fourier series, Sobolev spaces, and other prominent mathematical structures.