ABSTRACT
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 8.1 A Hot Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 8.2 The Hunt for a New Abstract Paradigm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 8.3 A Small Dose of Functional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
8.3.1 Moving Beyond Just RN and MN : Introducing the Notions of Banach Space and Hilbert Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 8.3.1.1 The Notion of Convergence Revisited . . . . . . . . . . . . . . . . . . 197 8.3.1.2 An Important Topological Notion-Closed Sets . . . . . . . . . 198
8.3.2 Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 8.3.3 Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 8.3.4 Calculus in Abstract Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
8.3.4.1 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 8.3.4.2 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 8.3.4.3 The Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 8.3.4.4 The Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
8.4 Looking Ahead . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
You were introduced to linear, nonhomogeneous, and semilinear systems of ODEs in RN in Part I. The theory developed there applied to an eclectic collection of contextually unrelated mathematical models. Alas, not all phenomena are described by a mathematical model that can be captured under this theoretical umbrella. There are various reasons for this, even within the realm of ODEs. The primary departure on which we shall now focus is the dependence of the unknown function (or “solution”) on more than one independent variable. As such, the equations used to form the mathematical models now involve partial derivatives, not ordinary ones. The purpose of Part II is to probe this issue more deeply and try to answer the question, “Can we somehow rewrite our mathematical models in a manner that resembles the Cauchy problems (HCP), (Non-CP) and (Semi-CP) from Part I?”
