Function Spaces and Hilbert Spaces

This lecture covers the concepts of function spaces and Hilbert spaces, starting with the definition of function spaces as vector spaces over C, whose elements are functions. It then introduces the structure of inner product spaces, focusing on the properties of scalar products. The lecture also discusses the importance of completeness in Hilbert spaces, where all Cauchy sequences converge. Examples of inner product spaces, such as the Coordinate space and the set of infinite square summable sequences, are presented. The lecture concludes by highlighting that finite-dimensional inner product spaces are Hilbert spaces, and that most properties of finite-dimensional spaces can be extended to infinite-dimensional Hilbert spaces.