ABSTRACT
This book covers the consequences of differentiation for functions of one variable and covers the basics leading up to extremal theory for functions of two variables. Along the way, a lot of attention is paid to developing the theory underlying these ideas properly. The number e is developed and all of its properties is developed from the sequence definition. Since ideas from sequences of functions and series are not yet known, all of the properties of the exponential and logarithm function must be explained using limit approaches. The theory of Riemann Integration and sequences and series of functions are explained in a careful manner. The book covers the basics of three new kinds of spaces: metric spaces, normed linear spaces and inner product spaces. It discusses the linear functional analysis along with Hahn — Banach Theorem, the open and closed mapping theorems and some spectral theory for linear operators of various kinds.
