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This monograph aims to give a self-contained introduction into the whole field of topological analysis: Requiring essentially only basic knowledge of elementary calculus and linear algebra, it provides all required background from topology, analysis, linear and nonlinear functional analysis, and multivalued maps, containing even basic topics like separation axioms, inverse and implicit function theorems, the Hahn-Banach theorem, Banach manifolds, or the most important concepts of continuity of multivalued maps. Thus, it can be used as additional material in basic courses on such topics. The main intention, however, is to provide also additional information on some fine points which are usually not discussed in such introductory courses. The selection of the topics is mainly motivated by the requirements for degree theory which is presented in various variants, starting from the elementary Brouwer degree (in Euclidean spaces and on manifolds) with several of its famous classical consequences, up to a general degree theory for function triples which applies for a large class of problems in a natural manner. Although it has been known to specialists that, in principle, such a general degree theory must exist, this is the first monograph in which the corresponding theory is developed in detail.

Topological degree. --- Topological spaces. --- Fredholm operators. --- Algebraic topology. --- Topology --- Operators, Fredholm --- Linear operators --- Spaces, Topological --- Degree, Topological --- Degree (Topology) --- Degree theory --- Algebraic topology --- Analysis. --- Banach Manifold. --- Fredholm. --- Hahn-Banach Theorem. --- Implicit Function Theorem. --- Inverse Function Theorem. --- Linear Functional Analysis. --- Multivalued Map. --- Nonlinear Functional Analysis. --- Nonlinear Inclusion. --- Separation Axiom. --- Topology. --- Triple Degree.

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In Hypo-Analytic Structures Franois Treves provides a systematic approach to the study of the differential structures on manifolds defined by systems of complex vector fields. Serving as his main examples are the elliptic complexes, among which the De Rham and Dolbeault are the best known, and the tangential Cauchy-Riemann operators. Basic geometric entities attached to those structures are isolated, such as maximally real submanifolds and orbits of the system. Treves discusses the existence, uniqueness, and approximation of local solutions to homogeneous and inhomogeneous equations and delimits their supports. The contents of this book consist of many results accumulated in the last decade by the author and his collaborators, but also include classical results, such as the Newlander-Nirenberg theorem. The reader will find an elementary description of the FBI transform, as well as examples of its use. Treves extends the main approximation and uniqueness results to first-order nonlinear equations by means of the Hamiltonian lift.Originally published in 1993.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Differential equations, Partial. --- Manifolds (Mathematics) --- Vector fields. --- Direction fields (Mathematics) --- Fields, Direction (Mathematics) --- Fields, Slope (Mathematics) --- Fields, Vector --- Slope fields (Mathematics) --- Vector analysis --- Geometry, Differential --- Topology --- Partial differential equations --- Algebra homomorphism. --- Analytic function. --- Automorphism. --- Basis (linear algebra). --- Bijection. --- Bounded operator. --- C0. --- CR manifold. --- Cauchy problem. --- Cauchy sequence. --- Cauchy–Riemann equations. --- Characterization (mathematics). --- Coefficient. --- Cohomology. --- Commutative property. --- Commutator. --- Complex dimension. --- Complex manifold. --- Complex number. --- Complex space. --- Complex-analytic variety. --- Continuous function (set theory). --- Corollary. --- Coset. --- De Rham cohomology. --- Diagram (category theory). --- Diffeomorphism. --- Differential form. --- Differential operator. --- Dimension (vector space). --- Dirac delta function. --- Dirac measure. --- Eigenvalues and eigenvectors. --- Embedding. --- Equation. --- Exact differential. --- Existential quantification. --- Exterior algebra. --- F-space. --- Formal power series. --- Frobenius theorem (differential topology). --- Frobenius theorem (real division algebras). --- H-vector. --- Hadamard three-circle theorem. --- Hahn–Banach theorem. --- Holomorphic function. --- Hypersurface. --- Hölder condition. --- Identity matrix. --- Infimum and supremum. --- Integer. --- Integral equation. --- Integral transform. --- Intersection (set theory). --- Jacobian matrix and determinant. --- Linear differential equation. --- Linear equation. --- Linear map. --- Lipschitz continuity. --- Manifold. --- Mean value theorem. --- Method of characteristics. --- Monomial. --- Multi-index notation. --- Neighbourhood (mathematics). --- Norm (mathematics). --- One-form. --- Open mapping theorem (complex analysis). --- Open mapping theorem. --- Open set. --- Ordinary differential equation. --- Partial differential equation. --- Poisson bracket. --- Polynomial. --- Power series. --- Projection (linear algebra). --- Pullback (category theory). --- Pullback (differential geometry). --- Pullback. --- Riemann mapping theorem. --- Riemann surface. --- Ring homomorphism. --- Sesquilinear form. --- Sobolev space. --- Special case. --- Stokes' theorem. --- Stone–Weierstrass theorem. --- Submanifold. --- Subset. --- Support (mathematics). --- Surjective function. --- Symplectic geometry. --- Symplectic vector space. --- Taylor series. --- Theorem. --- Unit disk. --- Upper half-plane. --- Vector bundle. --- Vector field. --- Volume form.