Some Remarks on Functional Analysis

Overview

Functional analysis is a key tool in the study of partial differential equations which helps to answer key questions such as existence, well-posedness, and the class in which a solution should belong. We begin these remarks by introducing normed spaces and Banach spaces and then bounded linear operators in normed spaces. Next, we define Hilbert spaces and consider aspects relating to linear operators on Hilbert spaces. With this structure, we are able to consider well-posed of problems and describe the notions of Hölder and Lyapunov stability and Hadamard well-posedness. Finally, we describe how some thermal stress problems can be formulated using abstract operator notation.

Introduction

Many of the theories involving thermal stresses, whether this be in thermoelasticity, in heat-conducting fluids, in viscoelastic materials, or in more exotic substances like auxetic foams, are based on partial differential equations (PDEs). To develop a model for a real-life situation, one...