Continuous Functions. Jacques Simon

      

      Table of Contents

      1  Cover

      2  Title Page

      3  Copyright Page

      4  Introduction

      5  Familiarization with Semi-normed Spaces

      6  Notations

      7  Chapter 1: Spaces of Continuous Functions 1.1 Notions of continuity 1.2 Spaces С(Ω; E), С_b(Ω; E), С_K(Ω; E), С(Ω; E) and С_b(Ω; E) 1.3 Comparison of spaces of continuous functions 1.4 Sequential completeness of spaces of continuous functions 1.5 Metrizability of spaces of continuous functions 1.6 The space

1.7 Continuous mappings 1.8 Continuous extension and restriction 1.9 Separation and permutation of variables 1.10 Sequential compactness in C_b(Ω; E)

      8  Chapter 2: Differentiable Functions 2.1 Differentiability 2.2 Finite increment theorem 2.3 Partial derivatives 2.4 Higher order partial derivatives 2.5 Spaces

and

2.6 Comparison and metrizability of spaces of differentiable functions 2.7. Filtering properties of spaces of differentiable functions 2.8. Sequential completeness of spaces of differentiable functions 2.9. The space

and the set

      9  Chapter 3: Differentiating Composite Functions and Others 3.1. Image under a linear mapping 3.2. Image under a multilinear mapping: Leibniz rule 3.3. Dual formula of the Leibniz rule 3.4. Continuity of the image under a multilinear mapping 3.5. Change of variables in a derivative 3.6. Differentiation with respect to a separated variable 3.7. Image under a differentiable mapping 3.8. Differentiation and translation 3.9. Localizing functions

      10  Chapter 4: Integrating Uniformly Continuous Functions 4.1. Measure of an open subset of

4.2. Integral of a uniformly continuous function 4.3. Case where E is not a Neumann space 4.4. Properties of the integral 4.5. Dependence of the integral on the domain of integration 4.6. Additivity with respect to the domain of integration 4.7. Continuity of the integral 4.8. Differentiating under the integral sign

      11  Chapter 5: Properties of the Measure of an Open Set 5.1. Additivity of the measure 5.2. Negligible sets 5.3. Determinant of d vectors 5.4. Measure of a parallelepiped

      12  Chapter 6: Additional Properties of the Integral 6.1. Contribution of a negligible set to the integral 6.2. Integration and differentiation in one dimension 6.3. Integration of a function of functions 6.4. Integrating a function of multiple variables 6.5. Integration between graphs 6.6. Integration by parts and weak vanishing condition for a function 6.7. Change of variables in an integral 6.8.