Differentiation and Integration

Although the continuous linear operators are of paramount importance as the morphisms of normed spaces, much can be deduced about ‘locally linear’ maps, otherwise known as differentiable functions. Particular consideration is given to generalizations of the Fundamental Theorem of Calculus, the Mean Value Theorem, and Taylor’s theorem to Banach spaces. The theory is applied to the Newton Raphson iteration method of solving equations. Complex differentiable functions, known as analytic functions, have distinctively more rigid properties, borne out by a string of results proved by Cauchy, including the Cauchy’s Residue theorem and Integral Formula.