Banach, Fréchet, Hilbert and Neumann Spaces

Jacques Simon, CNRS, France.

• Introduction xiFamiliarization with Semi-normed Spaces xvNotations xviiChapter 1 Prerequisites 11.1 Sets, mappings, orders 11.2 Countability 31.3 Construction of R 41.4 Properties of R 5Part 1 Semi-normed Spaces 9Chapter 2 Semi-normed Spaces 112.1 Definition of semi-normed spaces 112.2 Convergent sequences 152.3 Bounded, open and closed sets 172.4 Interior, closure, balls and semi-balls 212.5 Density, separability 232.6 Compact sets 252.7 Connected and convex sets 30Chapter 3 Comparison of Semi-normed Spaces 333.1 Equivalent families of semi-norms 333.2 Topological equalities and inclusions 343.3 Topological subspaces 393.4 Filtering families of semi-norms 433.5 Sums of sets 46Chapter 4 Banach, Fréchet and Neumann Spaces 494.1 Metrizable spaces 494.2 Properties of sets in metrizable spaces 514.3 Banach, Fréchet and Neumann spaces 554.4 Compacts sets in Fréchet spaces 574.5 Properties of R 584.6 Convergent sequences 604.7 Sequential completion of a semi-normed space 62Chapter 5 Hilbert Spaces 655.1 Hilbert spaces 655.2 Projection in a Hilbert space 685.3 The space Rd 70Chapter 6 Product, Intersection, Sum and Quotient of Spaces 736.1 Product of semi-normed spaces 736.2 Product of a semi-normed space by itself 786.3 Intersection of semi-normed spaces 806.4 Sum of semi-normed spaces 836.5 Direct sum of semi-normed spaces 896.6 Quotient space 93Part 2 Continuous Mappings 95Chapter 7 Continuous Mappings 977.1 Continuous mappings 977.2 Continuity and change of topology or restriction 1007.3 Continuity of composite mappings 1027.4 Continuous semi-norms 1027.5 Continuous linear mappings 1047.6 Continuous multilinear mappings 1077.7 Some continuous mappings 111Chapter 8 Images of Sets Under Continuous Mappings 1158.1 Images of open and closed sets 1158.2 Images of dense, separable and connected sets 1178.3 Images of compact sets 1198.4 Images under continuous linear mappings 1218.5 Continuous mappings in compact sets 1238.6 Continuous real mappings 1248.7 Compacting mappings 125Chapter 9 Properties of Mappings in Metrizable Spaces 1299.1 Continuous mappings in metrizable spaces 1299.2 Banach’s fixed point theorem 1339.3 Baire’s theorem 1349.4 Open mapping theorem 1369.5 Banach–Schauder’s continuity theorem 1389.6 Closed graph theorem 139Chapter 10 Extension of Mappings, Equicontinuity 14110.1 Extension of equalities by continuity 14110.2 Continuous extension of mappings 14210.3 Equicontinuous families of mappings 14610.4 Banach–Steinhaus equicontinuity theorem 148Chapter 11 Compactness in Mapping Spaces 15311.1 The spaces F(X; F) and C(X; F)-pt 15311.2 Zorn’s lemma 15411.3 Compactness in F(X; F) 15711.4 An Ascoli compactness theorem in C(X; F)-pt 161Chapter 12 Spaces of Linear or Multilinear Mappings 16312.1 The space L(E; F) 16312.2 Bounded sets in L(E; F) 16512.3 Sequential completeness of L(E; F) when E is metrizable 16712.4 Semi-norms and norm on L(E; F) when E isnormed 16912.5 Continuity of the composition of linear mappings 17112.6 Inversibility in the neighborhood of an isomorphism 17412.7 The space Ld(E1 × ··· × Ed; F) 17812.8 Separation of the variables of a multilinear mapping 181Part 3 Weak Topologies 187Chapter 13 Duality 18913.1 Dual 18913.2 Dual of a metrizable or normed space 19313.3 Dual of a Hilbert space 19613.4 Extraction of ∗ weakly converging subsequences 19913.5 Continuity of the bilinear form of duality 20313.6 Dual of a product 20513.7 Dual of a direct sum 206Chapter 14 Dual of a Subspace 20914.1 Hahn–Banach theorem 20914.2 Corollaries of the Hahn–Banach theorem 21114.3 Characterization of a dense subspace 21214.4 Dual of a subspace 21314.5 Dual of an intersection 21514.6 Dangerous identifications 216Chapter 15 Weak Topology 22115.1 Weak topology 22115.2 Weak continuity and topological inclusions 22415.3 Weak topology of a product 22515.4 Weak topology of an intersection 22615.5 Norm and semi-norms of a weak limit 228Chapter 16 Properties of Sets for the Weak Topology 23116.1 Banach–Mackey theorem (weakly bounded sets) 23116.2 Gauge of a convex open set 23316.3 Mazur’s theorem (weakly closed convex sets) 23516.4 ¢Smulian’s theorem (weakly compact sets) 23716.5 Semi-weak continuity of a bilinear mapping 240Chapter 17 Reflexivity 24317.1 Reflexive spaces 24317.2 Sequential completion of a semi-reflexive space 24717.3 Prereflexivity of metrizable spaces 24817.4 Reflexivity of Hilbert spaces 25017.5 Reflexivity of uniformly convex Banach spaces 25217.6 A property of the combinations of linear forms 25617.7 Characterizations of semi-reflexivity 25717.8 Reflexivity of a subspace 26117.9 Reflexivity of the image of a space 26117.10 Reflexivity of the dual 263Chapter 18 Extractable Spaces 26518.1 Extractable spaces 26518.2 Extractability of Hilbert spaces 26618.3 Extractability of semi-reflexive spaces 26718.4 Extractability of a subspace or of the image of a space 26918.5 Extractability of a product or of a sum of spaces 27018.6 Extractability of an intersection of spaces 27118.7 Sequential completion of extractable spaces 271Part 4 Differential Calculus 273Chapter 19 Differentiable Mappings 27519.1 Differentiable mappings 27519.2 Differentiality, continuity and linearity 27719.3 Differentiation and change of topology or restriction 27919.4 Mean value theorem 28119.5 Bounds on a real differentiable mapping 28419.6 Differentiation of a composite mapping 28619.7 Differential of an inverse mapping 28919.8 Inverse mapping theorem 290Chapter 20 Differentiation of Multivariable Mappings 29520.1 Partial differentiation 29520.2 Differentiation of a multilinear or multi-component mapping 29820.3 Differentiation of a composite multilinear mapping 300Chapter 21 Successive Differentiations 30321.1 Successive differentiations 30321.2 Schwarz’s symmetry principle 30521.3 Successive differentiations of a composite mapping 308Chapter 22 Derivation of Functions of One Real Variable 31322.1 Derivative of a function of one real variable 31322.2 Derivative of a real function of one real variable 31522.3 Leibniz formula 31922.4 Derivatives of the power, logarithm and exponential functions 320Bibliography 325Cited Authors 331Index 335