# Book:Philippe G. Ciarlet/Linear and Nonlinear Functional Analysis with Applications

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## Philippe G. Ciarlet:

## Philippe G. Ciarlet: *Linear and Nonlinear Functional Analysis with Applications*

Published $2013$, **Society for Industrial and Applied Mathematics**

- ISBN 978-1611972580.

### Subject Matter

### Contents

**Preface**

**1. Real Analysis and Theory of Functions: A Quick Review**

- Introduction

- 1.1 Sets

- 1.2 Mappings

- 1.3 The axiom of choice and Zorn's lemma

- 1.4 Construction of the sets $\R$ and $\C$

- 1.5 Cardinal numbers; finite and infinite sets

- 1.6 Topological spaces

- 1.7 Continuity in topological spaces

- 1.8 Compactness in topological spaces

- 1.9 Conectedness and simple-conectedness in topological spaces

- 1.10 Metric spaces

- 1.11 Continuity and uniform continuity in metric spaces

- 1.12 Complete metric spaces

- 1.13 Compactness in metric spaces

- 1.14 The Lebesgue measure in $\R^n$; measurable functions

- 1.15 The Lebesgue integral in $\R^n$; the basic theorems

- 1.16 Change of variable in Lebesgue integrals in $\R^n$

- 1.17 Volumes, areas, and lengths in $\R^n$

- 1.18 The spaces $\map {\cal {C}^m} \Omega$ and $\map {\cal {C}^m} {\bar \Omega}$; domains in $\R^n$

**2. Normed Vector Spaces**

- Introduction

- 2.1 Vector spaces; Hamel bases; dimension of a vector space

- 2.2 Normed vector spaces; first properties and examples; quotient spaces

- 2.3 The space $\map {\cal C} {K;Y}$ with $K$ compact; uniform convergence and local uniform convergence

- 2.4 The spaces ${\cal l}^p, 1 \le p \le \infty$

- 2.5 The Lebesgue spaces $\map {L^p} \Omega, 1 \le p \le \infty$

- 2.6 Regularization and approximation in the spaces $\map {L^p} \Omega, 1 \le p < \infty$

- 2.7 Compactness and finite-dimensional normed vector spaces; F. Riesz theorem

- 2.8 Application of compactness in finite-dimensional normed vector space: The fundamental theorem of algebra

- 2.9 Continuous linear operators in normed vector soaces; the spaces $\map {\cal L} {X;Y}$, $\map {\cal L} X$, and $X'$

- 2.10 Compact linear operatros in normed vector spaces

- 2.11 Continuous multilinear mappings in normed vector spaces; the space $\map { {\cal L}_k} {X_1, X_2, \dots , X_k; Y}$

- 2.12 Korovkin's theorem

- 2.13 Application of Korovkin's theorem to polynomial approximation; Bohman's, Bernstein's, and Weierstrass' theorems

- 2.14 Application of Korovkin's theorem to trigonometric polynomial approximation; Fejér's theorem

- 2.15 The Stone-Weierstrass theorem

- 2.16 Convex sets

- 2.17 Convex functions

**3. Banach Spaces**

- Introduction

- 3.1 Banach spaces; first properties

- 3.2 First examples of Banach spaces; the spaces $\map {\cal C} {K;Y}$ with $K$ compact and $Y$ complete, and ${\cal L} {X;Y}$ with $Y$ complete

- 3.3 Integral of a continuous function of a real variable with values in a Banach space

- 3.4 Further examples of Banach spaces: the spaces $l^p$ and $\map {L^p} \Omega$, $, \le p \le \infty$

- 3.5 Dual of a normed vector space; first examples; F. Riesz representation theorem in $\map {L^p} \Omega$, $1 \le p < \infty$

- 3.6 Series in Banach space

- 3.7 Banach fixed point theorem

- 3.8 Application of Banach fixed point theorem: Existence of solutions to nonlinear ordinary differential equations; Cauchy-Lipschitz theorem; the pendulum equation

- 3.9 Applictaion of Banach fixed point theorem: Existence of solutions to nonlinear two-point boundary value problems

- 3.10 Ascoli-Arzelà's theorem

- 3.11 Application of Ascoli-Arzelà's theorem: Existence of solutions to nonlinear ordinary differential equations; Cauchy-Peano theorem; Euler's method

**4. Inner Product Spaces and Hilbert Spaces**

- Introduction

- 4.1 Inner-product spaces and Hilbert spaces; first properties; Cauchy-Schwarz-Bunyakovskii inequality; parallelogram law

- 4.2 First examples of inner-product spaces and Hilbert spaces; the spaces $l^2$ and $\map {L^2} \Omega$

- 4.3 The projection theorem

- 4.4 Application of the projection theorem; Least-squares solution of a linear system

- 4.5 Orthogonality; direct sum theorem

- 4.6 F. Riesz representation theorem in a Hilbert space

- 4.7 First applications of the F. Riesz representation theorem: Hahn-Banach theorem in a Hilbert space; adjoint operators; reproducing kernels

- 4.8 Maximal orthonormal families in an inner-product space

- 4.9 Hilbert bases and Fourier series in a Hilbert space

- 4.10 Eigenvalues and eigenvectors of self-adjoint operators in inner-product spaces

- 4.11 The spectral theorem for compact self-adjoint operators

**5. The "Great Theorems" of Linear Functional Analysis**

- Introduction

- 5.1 Baire's theorem; a first application: Noncompleteness of the space of all polynomials

- 5.2 Application of Baire's theorem: Existence of nowhere differentiable continuous functions

- 5.3 Banach-Steinhaus theorem, alias the uniform boundedness principle; application to numerical quadrature formulas

- 5.4 Application of the Banach-Steinhaus theorem: Divergence of Lagrange interpolation

- 5.5 Application of the Banach-Steinhaus theorem: Divergence of Fourier series

- 5.6 Banach open mapping theorem; a first application: Well-posedness of two-point boundary value problems

- 5.7 Banach closed graph theorem; a first application: Hellinger-Toeplitz theorem

- 5.8 The Hahn-Banach theorem in a vector space

- 5.9 The Hahn-Banach theorem in a normed vector space: first consequences

- 5.10 Geometric forms of the Hahn-Banach theorem; separation of convex sets

- 5.11 Dual operators; Banach closed range theorem

- 5.12 Weak convergence and weak* convergence

- 5.13 Banach-Saks-Mazur theorem

- 5.14 Reflexive spaces; the Banach-Eberlein-Šmulian theorem

**6. Linear Partial Differential Equations**

- Introduction

- 6.1 Quadratic minimization problems; variational equations and variational inequalities

- 6.2 The Max-Milgram lemma

- 6.3 Weak partial derivative in $\map {L_{loc}^1} \Omega$; a brief incursion into distribution theory

- 6.4 Hypoellipticity of $\Delta$

- 6.5 The Sobolev spaces $\map {W^{m,p} } \Omega$ and $\map {H^m} \Omega$: First properties

- 6.6 The Sobolev spaces $\map {W^{m,p} } \Omega$ and $\map {H^m} \Omega$ with $\Omega$ a domain: imbedding theorems, traces, Green's formulas

- 6.7 Examples of second-order linear elliptic boundary value problems; the membrane problem

- 6.8 Examples of fourth-order linear boundary value problems; the biharmonic and plate problems

- 6.9 Examples of nonlinear boundary value problems associated with variational inequalities; obstacle problems

- 6.10 Eigenvalue problems for second-order elliptic operators

- 6.11 The spaces $\map {W^{-m,q} } \Omega$ and $\map {H^{-m} } \Omega$; J. L. Lions lemma

- 6.12 The Babuška-Brezzi inf-sup theorem; application to constrained quadratic minimization problems

- 6.13 Application of the Babuška-Brezzi inf-sup theorem: Primal, mixed, and dual formulations of variational problems

- 6.14 Application of the Babuška-Brezzi inf-sup theorem and of J. L. Lions lemma: The Stokes equations

- 6.15 A second application of J. L. Lions lemma: Korn's inequalitity

- 6.16 Application of Korn's inequality: The equations of three-dimensional linearized elasticity

- 6.17 The classical Poincaré lemma and its weak version as an application of J. L. Lions lemma and of the hypoellipticity of $\Delta$

- 6.18 Application of Poincaré's lemma: The classical and weak Saint-Venant lemmas; the Cesàro-Volterra path integral formula

- 6.19 Another application of J. L. Lions lemma: The Donati lemmas

- 6.20 Pfaff systems

**7. Differential Calculus in Normed Vector Spaces**

- Introduction

- 7.1 The Fréchet derivative; the chain rule; the Piola identity; application to extrema of real-valued functions

- 7.2 The mean value theorem in a normed vector space; first applications

- 7.3 Application of the mean value theorem: Differentiability of the limit of a sequence of differentiable functions

- 7.4 Application of the mean value theorem: Differentiability of a function defined by an integral

- 7.5 Application of the mean value theorem: Sard's theorem

- 7.6 A mean value theorem for functions of class ${\cal C}^1$ with values in a Banach space

- 7.7 Newton's method for solving nonlinear equations; the Newton-Kantorovich theorem in a Banach space

- 7.8 Higher order derivatives; Schwarz lemma

- 7.9 Taylor formulas; application to extrema of real-valued functions

- 7.10 Application: Maximum principle for second-order linear elliptic operators

- 7.11 Application: Lagrange interpolation in $\R^n$ and multipoint Taylor formulas

- 7.12 Convex functions and differentiability; application to extrema of real-valued functions

- 7.13 The implicit functions theorem: first application: Class ${\cal C}^\infty$ of the mapping $A \rightarrow A^{-1}$

- 7.14 The local inversion theorem; the invariance of domain theorem for mappings of class ${\cal C}^1$ in Banach spaces; class ${\cal C}^\infty$ of the mapping $A \rightarrow A^{1/2}$

- 7.15 Constrained extrema of real-valued functions; Lagrange multipliers

- 7.16 Lagrangians and saddle-points; primal and dual problems

**8. Differential Geometry in $\R^n$**

- Introduction

- 8.1 Curvilinear coordinates in an open subset of $\R^n$

- 8.2 Metric tensor; volumes and legths in curvilinear coordinates

- 8.3 Covariant derivative of a vector field

- 8.4 Tensors - a brief introduction

- 8.5 Necessary conditions satisfied by the metric tensor; the Riemann curvature tensor

- 8.6 Existence of an immersion on an open subset $\R^n$ with a prescribed metric tensor; the fundamental theorem of Riemannian geometry

- 8.7 Uniqueness up to isometries of immersions with the same metric tensor; the rigidity theorem for an open subset of $\R^n$

- 8.8 Curvilinear coordinates on a surface in $\R^3$

- 8.9 First fundamental form of a surface; areas, lengths, and angles on a surface

- 8.10 Isometric, equiareal, and conformal surfaces

- 8.11 Second fundamental form of a surface; curvature on a surface

- 8.12 Principal curvatures; Gaussian curvature

- 8.13 Covariant derivatives of a vector field defined on a surface; the Gauss and Weingarten formulas

- 8.14 Necessary conditions satisfied by the first and second fundamental forms; The Gauss and Codazzi-Mainardi equations

- 8.15 Gauss Theorema Egregium; applicaiton to cartography

- 8.16 Existence of a surface with prescribed first and second fundamental forms; the fundamental theorem of surface theory

- 8.17 Uniqueness of surfaces with the same fundamental forms; the rigidity theorem for surfaces

**9. The "Great Theorems" of Nonlinear Functional Analysis**

- Introduction

- 9.1 Nonlinear partial differential equations as the Euler-Lagrange equations associated with the minimization of a functional

- 9.2 Convex function and sequentially lower semicontinuous functions with values in $\R \cup \set \infty$

- 9.3 Existence of minimizers for coercive and sequentailly weakly lower semicontinuous functionals

- 9.4 Applications to the von Kármán equations

- 9.5 Existence of minimizers in $\map {W^{1,p} } \Omega$

- 9.6 Application to the $p-$Laplace operator

- 9.7 Polyconvexity; compensated compactness; John Ball's existence theorem in nonlinear elasticity

- 9.8 Ekeland's variational principle; existence of minimizers for functional that satisfy the Palais-Smale condition

- 9.9 Brouwer's fixed point theorem - a first proof

- 9.10 Application of Brouwer's theorem to the von Kármán equations, by means of the Galerkin method

- 9.11 Application of Brouwer's theorem to the Navier-Stokes equations, by means of the Galerkin method

- 9.12 Schauder's fixed point theorem; Schäfer's fixed point theorem; Leray-Schauder fixed point theorem

- 9.13 Monotone operators

- 9.14 The Minty-Browder theorem for monotone operators; application to the $p-$Laplace operator

- 9.15 The Brouwer topological degree in $\R^n$: Definition and properties

- 9.16 Brouwer's fixed point theorem - a second proof - and the hairy ball theorem

- 9.17 Borsuk's and Borsuk-Ulam theorems; Brouwer's invariance of domain theorem

**Bibliographical Notes**

**Bibliography**

**Main Notations**

**Index**