(An) integral representation theorem for topological vector spaces and the spectral theorem

Abstract

The Riesz Representation Theorem is the theorem that represents a continuous linear operator from C(S) into scalars as an integral form. In this thesis, we generalized the Riesz Representation Theorem for the case that scalars are changed into a locally convex Hausdorff topological vector space. And prove the Spectral Theorem for bounded self-adjoint operators on a complex Hilbert Space using the integral representation theorem.