Continuous Functions

This book is the second of a set dedicated to the mathematical tools used in partial differential equations derived from physics. It presents the properties of continuous functions, which are useful for solving partial differential equations, and, more particularly, for constructing distributions valued in a Neumann space. The author examines partial derivatives, the construction of primitives, integration and the weighting of value functions in a Neumann space. Many of them are new generalizations of classical properties for values in a Banach space. Simple methods, semi-norms, sequential properties and others are discussed, making these tools accessible to the greatest number of students – doctoral students, postgraduate students – engineers and researchers, without restricting or generalizing the results.

Introduction ix Familiarization with Semi-normed Spaces xiii Notations xv Chapter 1. Spaces of Continuous Functions 1 1.1. Notions of continuity 1 1.2. Spaces C(Ω;E), Cb(Ω;E), CK(Ω;E), C(Ω;E) and Cb(Ω;E) 3 1.3. Comparison of spaces of continuous functions 6 1.4. Sequential completeness of spaces of continuous functions 10 1.5. Metrizability of spaces of continuous functions 11 1.6. The space K(Ω;E) 14 1.7. Continuous mappings 20 1.8. Continuous extension and restriction 22 1.9. Separation and permutation of variables 23 1.10. Sequential compactness in Cb(Ω;E) 28 Chapter 2. Differentiable Functions 31 2.1. Differentiability 31 2.2. Finite increment theorem 34 2.3. Partial derivatives 37 2.4. Higher order partial derivatives 40 2.5. Spaces Cm(Ω;E), Cmb(Ω;E), CmK