Riesz representation theorem

Abstract

This master's report presents a proof of the Riesz Representation Theorem for Hilbert spaces. The proof is developed from the basic geometry of inner product spaces, beginning with the Cauchy-Bunyakowski-Schwartz Inequality, the norm induced by an inner product, and the Parallelogram Law, and then using the Closest Point Theorem and the Orthogonal Projection Theorem. The report also discusses two applications of the Riesz Representation Theorem: the existence and basic properties of adjoint operators, and a special case of the Radon-Nikodym Theorem for finite measure spaces in which one measure is bounded above by a constant multiple of another.