Distributions

Introduction

Objective. This book is the third of seven volumes dedicated to solving partial differential equations in physics:

Volume 1: Banach, Frechet, Hilbert and Neumann Spaces

Volume 2: Continuous Functions

Volume 3: Distributions

Volume 4: Integration

Volume 5: Sobolev Spaces

Volume 6: Traces

Volume 7: Partial Differential equations

This third volume aims to construct the space of distributions with real or vectorial values and to provide the main properties that are useful in studying partial differential equations.

Intended audience. We¹ have looked for simple methods that require a minimal level of knowledge to make this tool accessible to as wide an audience as possible — doctoral students, university students, engineers — without loosing generality and even generalizing certain results, which may be of interest to some researchers.

This has led us to choose an unconventional approach that prioritizes semi-norms and sequential properties, whether related to completeness, compactness or continuity.

Utility of distributions. The main advantage of distributions is that they provide derivatives of all continuous or integrable functions, even those which are not differentiable, and thus broaden the scope of application of differential calculus. This is especially useful for solving partial differential equations.

To this end, a family of objects, the distributions, is defined, with the following properties.

• — Any continuous function is a distribution.

• — Any distribution ...