Linear and Nonlinear Functional Analysis with Applications

✍ Scribed by Philippe G. Ciarlet

• Publisher: SIAM-Society for Industrial and Applied Mathematics
• Year: 2013
• Tongue: English
• Leaves: 849
• Category: Library

✦ Synopsis

This single-volume textbook covers the fundamentals of linear and nonlinear functional analysis, illustrating most of the basic theorems with numerous applications to linear and nonlinear partial differential equations and to selected topics from numerical analysis and optimization theory.

This book has pedagogical appeal because it features self-contained and complete proofs of most of the theorems, some of which are not always easy to locate in the literature or are difficult to reconstitute. It also offers 401 problems and 52 figures, plus historical notes and many original references that provide an idea of the genesis of the important results, and it covers most of the core topics from functional analysis.

Audience: Linear and Nonlinear Functional Analysis with Applications is intended for advanced undergraduates, graduate students, and researchers and is ideal for teaching or self-study.

Contents: Preface; Chapter 1: Real analysis and theory of functions: A quick review; Chapter 2: Normed vector spaces; Chapter 3: Banach spaces; Chapter 4: Inner-product spaces and Hilbert spaces; Chapter 5: The great theorems of linear functional analysis; Chapter 6: Linear partial differential equations; Chapter 7: Differential calculus in normed vector spaces; Chapter 8: Differential geometry in Rn; Chapter 9: The great theorems of nonlinear functional analysis; Bibliographical notes; Bibliography; Main notations; Index.

✦ Table of Contents

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 Connectedness and simple-connectedness 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 ℝⁿ; measurable functions
1.15 The Lebesgue integral in ℝⁿ; the basic theorems
1.16 Change of variable in Lebesgue integrals in ℝⁿ
1.17 Volumes, areas, and lengths in ℝⁿ
1.18 The spaces 𝑪ᵐ(Ω) and 𝑪ᵐ(Ω); domains in ℝⁿ
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 𝑪(K; Y) with K compact; uniform convergence and local uniform convergence
2.4 The spaces 𝓵ᵖ, 1 < p < ∞
2.5 The Lebesgue spaces 𝑳ᵖ(Ω), 1 < p < ∞
2.6 Regularization and approximation in the spaces 𝑳ᵖ(Ω), 1 < p < ∞
2.7 Compactness and finite-dimensional normed vector spaces; F. Riesz theorem
2.8 Application of compactness in finite-dimensional normed vector spaces: The fundamental theorem of algebra
2.9 Continuous linear operators in normed vector spaces; the spaces 𝓛(X; Y), 𝓛(X), and X'
2.10 Compact linear operators in normed vector spaces
2.11 Continuous multilinear mappings in normed vector spaces; the space 𝓛ₖ (X₁, X₂, . . . , Xₖ; Y)
2.12 Korovkin's theorem
2.13 Application of Korovkin's theorem to polynomial approximation; Bohman's, Bernstein's, and Weierstrass's theorems
2.14 Application of Korovkin's theorem to trigonometric polynomial approximation; Fejer'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 𝑪(K; Y) with K compact and Y complete, and 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 𝓵ᵖ and 𝑳ᵖ(Ω), 1 < p < ∞
3.5 Dual of a normed vector space; first examples; F. Riesz representation theorem in 𝑳ᵖ(Ω), 1 < p < ∞
3.6 Series in Banach spaces
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 Application of Banach fixed point theorem: Existence of solutions to nonlinear two-point boundary value problems
3.10 Ascoli-Arzela's theorem
3.11 Application of Ascoli-Arzela'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 𝓵² and 𝑳²(Ω)
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-Smulian theorem
6 Linear Partial Differential Equations
Introduction
6.1 Quadratic minimization problems; variational equations and variational inequalities
6.2 The Lax-Milgram lemma
6.3 Weak partial derivatives in L¹loc(Ω); a brief incursion into distribution theory
6.4 Hypoellipticity of Δ
6.5 The Sobolev spaces 𝑾 m,p(Ω) and 𝑯 m(Ω): First properties
6.6 The Sobolev spaces 𝑾 m,p(Ω) and 𝑯 m(Ω) with Ω, 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 𝑾 -m,q(Ω) and 𝑯 -m(Ω); J.L. Lions lemma
6.12 The Babuska-Brezzi inf-sup theorem; application to constrained quadratic minimization problems
6.13 Application of the Babuska-Brezzi inf-sup theorem: Primal, mixed, and dual formulations of variational problems
6.14 Application of the Babuska-Brezzi inf-sup theorem and of J.L. Lions lemma: rrhe Stokes equations
6.15 A second application of J .L. Lions lemma: Korn's inequality
6.16 Application of Korn's inequality: The equations of three-dimensional linearized elasticity
6.17 The classical Poincare lemma and its weak version as an application of J .L. Lions lemma and of the hypoellipticity of Δ
6.18 Application of Poincare's lemma: The classical and weak Saint- Venant lemmas; the Cesaro-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 Frechet 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 𝑪¹ 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 ℝⁿ and multipoint Taylor formulas
7.12 Convex functions and differentiability; application to extrema of real-valued functions
7.13 The implicit function theorem; first application: Class 𝑪∞ of the mapping A 🡢 A⁻¹
7.14 The local inversion theorem; the invariance of domain theorem for mappings of class 𝑪¹ in Banach spaces; class 𝑪∞ of the mapping A 🡢 A¹ᐟ²
7.15 Constrained extrema of real-valued functions; Lagrange multipliers
7.16 Lagrangians and saddle-points; primal and dual problems
8 Differential Geometry in ℝⁿ
Introduction
8.1 Curvilinear coordinates in an open subset of ℝⁿ
8.2 Metric tensor; volumes and lengths 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 of ℝⁿ 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 ℝⁿ
8.8 Curvilinear coordinates on a surface in ℝ³
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; application 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 functions and sequentially lower semicontinuous functions with values in ℝ∪{∞}
9.3 Existence of minimizers for coercive and sequentially weakly lower semicontinuous functionals
9.4 Application to the von Karman equations
9.5 Existence of minimizers in 𝑾 1,p(Ω)
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 functionals 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 Karman 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; Schafer'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 ℝⁿ: 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