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

Subject Matter

 * Functional Analysis

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


 * 7. Differential Calculus in Normed Vector Spaces


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


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

Bibliographical Notes

Bibliography

Main Notations

Index