This book can be read independently by readers with a basic knowledge in general topology and normed vector spaces including the operator norm of a bounded linear operator To set the stage properly we review briefly the necessary elements of metric topology also Proofs are developed in detail with all steps justified Almost all previously proven results used within proofs and solutions to examples are explicitly cited and referred to by number This eliminates unnecessary time and frustration spent figuring out exactly which results were used and where they were first proved This book is used for a course at the post-graduate and honours graduate level It is my hope that the books will also appeal to university students in mathematicsA fundamental problem in applications of functional analysis is connected with the construction of suitable spaces to handle a given application Often the spaces needed are complete normed vector spaces Banach spaces constructed from spaces of continuous functions In this spirit slightly novel construction of the Lp-spaces together with proofs of the basic inequalities of holder and Monkowski in these spaces are also incorporated in this book Other important Banach spaces arising from function spaces and sequence spaces are also treated in this bookThe main bulk of the book presents the basic elements of the theory of Hilbert spaces and operators on Hilbert spaces culminating in a proof of the spectral theorem for compact self-adjoint operators on separable Hilbert spaces Notions from general topology and basic elements of the theory of normed vector spaces are in general used throughout this book without further notice