赋范向量空间上的微积分

Preface1 Normed Vector Spaces1．1 First Notions1．2 Limits and Continuity1．3 Open and Closed Sets1．4 Compactness1．5 Banach Spaces1．6 Linear and Polynomial Mappings1．7 Normed Algebras1．8 The Exponential MappingAppendix： The Fundamental Theorem of Algebra2 Differentiation2．1 Directional Derivatives2．2 The Differential2．3 Differentials of Compositions2．4 Mappings of Class C12．5 Extrema2．6 Differentiability of the NormAppendix： Gateaux Differentiability3 Mean Value Theorems3．1 Generalizing the Mean Value Theorem3．2 Partial Differentials3．3 Integration3．4 Differentiation under the Integral Sign4 Higher Derivatives and Differentials4．1 Schwarz's Theorem4．2 Operationson Ck-Mappings4．3 Multilinear Mappings4．4 Higher Differentials4．5 Higher Differentials and Higher Derivatives4．6 Cartesian Product Image Spaces4．7 Higher Partial Differentials4．8 Generalizing Ck to Normed Vector Spaces4．9 Leibniz's Rule5 Taylor Theorems and Applicahons5．1 Taylor Formulas5．2 Asymptotic Developments5．3 Extrema： Second-Order ConditionsAppendix： Homogeneous Polynomials6 Hilbert Spaces6．1 Basic Notions6．2 Projections6．3 The Distance Mapping6．4 The Riesz Representation Theorem7 Convex Functions7．1 Preliminary Results7．2 Continuity of Convex Functions7．3 Differentiable Convex Functions7．4 Extrema of Convex FunctionsAppendix： Convex Polyhedra8 The Inverse and Implicit Mapping Theorems8．1 The Inverse Mapping Theorem8．2 The Implicit Mapping Theorem8．3 The Rank Theorem8．4 Constrained ExtremaAppendix 1： Bijective Continuous Linear MappingsAppendix 2： Contractions9 Vector Fields9．1 Existence of Integral Curves9．2 Initial Conditions9．3 Geometrical Properties of Integral Curves9．4 Complete Vector FieldsAppendix： A Useful Result on Smooth Functions10 The Flow of a Vector Field10．1 Continuity of the Flow10．2 Differentiability of the Flow10．3 Higher Differentiability of the Flow10．4 The Reduced Flow10．5 One-Parameter Subgroups11 The Calculus of Variations： An Introduction11．1 The Space C1（I，E）11．2 Lagrangian Mappings11．3 Fixed Endpoint Problems11．4 Euler-LagrangeEquations11．5 Convexity11．6 The Class of an Extremal ReferencesIndex