Continuous Functions

Оглавление

Jacques Simon. Continuous Functions

Table of Contents

List of Illustrations

Guide

Pages

Volume 2. Continuous Functions

Introduction

Notes

Familiarization with Semi-normed Spaces

Notations. SPACES OF FUNCTIONS

OPERATIONS ON A FUNCTION f

DERIVATIVES OF A FUNCTION f

INTEGRALS AND PATHS

SEPARATED SEMI-NORMED SPACES

POINTS AND SETS IN ℝd

OTHER SETS

SPECIAL FUNCTIONS

TYPOGRAPHY

Chapter 1. Spaces of Continuous Functions

1.1 Notions of continuity

1.2 Spaces С(Ω; E), Сb(Ω; E), СK(Ω; E), С(Ω; E) and Сb(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 Cb(Ω; E)

Notes

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

Notes

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

Notes

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

Notes

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

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. Some particular changes of variables in an integral

Notes

Chapter 7. Weighting and Regularization of Functions

7.1. Weighting

7.2. Properties of weighting

7.3. Weighting of differentiable functions

7.4. Local regularization

7.5. Global regularization

7.6. Partition of unity

7.7. Separability of K∞(Ω)

Notes

Chapter 8. Line Integral of a Vector Field Along a Path

8.1. Paths

8.2. Line integral of a field along a path

8.3. Line integral along a concatenation of paths

8.4. Tubular flow and the concentration theorem

8.5. Invariance under homotopy of the line integral of a local gradient

Notes

Chapter 9. Primitives of Continuous Functions

9.1. Explicit primitive of a field with line integral zero

9.2. Primitive of a field orthogonal to the divergence-free test fields

9.3. Gluing of local primitives on a simply connected open set

9.4. Explicit primitive on a star-shaped set: Poincaré’s theorem

9.5. Explicit primitive under the weak Poincaré condition

9.6. Primitives on a simply connected open set

9.7. Comparison of the existence conditions for a primitive

9.8. Fields with local primitives but no global primitive

9.9. Uniqueness of primitives

9.10. Continuous primitive mapping

Notes

Chapter 10. Additional Results: Integration on a Sphere

10.1. Surface integration on a sphere

10.2. Properties of the integral on a sphere

10.3. Radial calculation of integrals

10.4. Surface integral as an integral of dimension d − 1

10.5. A Stokes formula

Notes

Appendix. Reminders

A.1. Notation and numbering

A.2. Semi-normed spaces. Vector spaces and semi-norms

Open, closed and bounded sets, sequences, connectedness

Compactness

The space ℝd, metrizable spaces, Fréchet spaces and Neumann spaces

A.3. Continuous mappings and duality. Continuous mappings

Linear or multilinear mappings

Dual and weak topology

A.4. Differentiable mappings and differentiable functions

Notes

Bibliography

Index. A

B

C

D

E

F

H

I

J, K

L

M

N

O

P

R

S

T

U

V, W

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Отрывок из книги

To Claire and Patricia, By your gaiety, “joie de vivre”, and femininity, you have embellished my life, and you have allowed me to conserve the tenacity needed for this endeavor

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— The concentration theorem for the integral and the construction of an incompressible tubular flow (Theorems 8.18 and 8.17), which are key steps in our construction of the primitives of a field taking values in a Neumann space, as it is explained in the comment Utility of the concentration theorem, p. 186.

Prerequisites. The proofs in the main body of the text only use definitions and results established in Volume 1, whose statements are recalled either in the text or in the Appendix. Detailed proofs are given, including arguments that may seem trivial to experienced readers, and the theorem numbers are systematically referenced.

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