Applied Functional Analysis
John Wiley & Sons, Feb 4, 2000 - Mathematics - 495 pages
A novel, practical introduction to functional analysis
In the twenty years since the first edition of Applied Functional Analysis was published, there has been an explosion in the number of books on functional analysis. Yet none of these offers the unique perspective of this new edition. Jean-Pierre Aubin updates his popular reference on functional analysis with new insights and recent discoveries-adding three new chapters on set-valued analysis and convex analysis, viability kernels and capture basins, and first-order partial differential equations. He presents, for the first time at an introductory level, the extension of differential calculus in the framework of both the theory of distributions and set-valued analysis, and discusses their application for studying boundary-value problems for elliptic and parabolic partial differential equations and for systems of first-order partial differential equations.
To keep the presentation concise and accessible, Jean-Pierre Aubin introduces functional analysis through the simple Hilbertian structure. He seamlessly blends pure mathematics with applied areas that illustrate the theory, incorporating a broad range of examples from numerical analysis, systems theory, calculus of variations, control and optimization theory, convex and nonsmooth analysis, and more. Finally, a summary of the essential theorems as well as exercises reinforcing key concepts are provided. Applied Functional Analysis, Second Edition is an excellent and timely resource for both pure and applied mathematicians.
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A Guide to the Reader
Theorems on Extension and Separation
Dual Spaces and Transposed Operators
The Banach Theorem and the BanachSteinhaus Theorem
Construction of Hilbert Spaces
Some Approximation Procedures in Spaces of Functions
Sobolev Spaces of Functions of Several Variables and the Fourier
Introduction to SetValued Analysis and Convex Analysis
Boundary Value Problems
DifferentialOperational Equations and Semigroups of Operators
Viability Kernels and Capture Basins
FirstOrder Partial Differential Equations
Selection of Results
Elementary Spectral Theory
HilbertSchmidt Operators and Tensor Products
Other editions - View all
associate backward invariant belongs bilinear mapping boundary value problems bounded capture basin Cauchy sequence Chapter characterize closed subset compact operator compact support Consequently contingent cone continuous linear form continuous linear operator convex function deduce defined Definition denned denote derivative distributions Dom(u Dom(v domain dual duality operator epiderivative epigraph equal finite dimensional Fourier transform graph Green's formula Hence Hilbert space Hilbert-Schmidt operator Hm(Q Hm(U implies inequality injective isometry isomorphism Lemma Lipschitz lower semicontinuous minimal Moreover nonempty nontrivial norm obtain orthogonal projector orthonormal base partial differential equations pivot space pointwise polynomials pre-Hilbert space Proof properties Proposition quotients Remark satisfying scalar product Section semigroup sequence of elements set-valued map Sobolev spaces space F subdifferential Suppose surjective tensor product transpose unbounded operator unique extension vector space vector subspace viability kernel
Page xi - This is quite natural, though, because each problem demands its own amount of properties that the derivative should enjoy (ie, its own degree of regularity). Without going too far by always requiring minimal assumptions, some problems could not be solved by sticking to the richest structure. The right balance between generality and readability is naturally a subjective choice.