Direct and projective limits of geometric Banach structures. (2024)

• Record Type: Book
• Further Information: Note: Patrick Cabau, Fernand Pelletier, with the assistance and partial collaboration of Daniel Beliţă.
• Authors: Cabau, Patrick; Pelletier, Fernand
• Other Names: Beliţă, Daniel collaborator.
• Contents: Acknowledgements Preface Author List of Figures 1. Preliminaries; 1.1 Banach spaces; 1.2 Examples of Banach spaces; 1.3 Dual spaces; 1.4 Properties; 1.5 Derivatives in Banach spaces; 1.6 Ordinary differential equations; 1.7 Banach manifolds; 1.8 Banach manifold structures on sets of maps; 1.9 Banach submanifolds; 1.10 Banach-Lie groups and Lie algebras; 1.11 Banach vector bundles; 1.12 Jets of sections of a vector bundle; 1.13 Notes 2. Banach Lie structures <br />2.1 Linear tensor structures; 2.2 Banach $G$-structures and tensor structures; 2.3 Examples of tensor structures on a Banach bundle; 2.4 Examples of integrable tensor structures on a Banach manifold; 2.5 Darboux Theorem for symplectic forms on a Banach manifold; 2.6 Notes; ; 3. Convenient structures; 3.1 Locally convex topological vector spaces; 3.2 Differential calculus on locally convex topological vector spaces; 3.3 Convenient calculus; 3.4 Convenient manifolds; 3.5 Tangent vectors; 3.6 Tangent mappings; 3.7 Immersions and submersions; 3.8 Convenient submanifolds; 3.9 Convenient bundles; 3.10 Convenient vector bundles; 3.11 Convenient tangent bundles; 3.12 Convenient vector fields; 3.13 Convenient cotangent bundles; 3.14 Convenient differential forms; 3.15 Connections on a convenient bundle; 3.16 Convenient Lie groups and Lie algebras; 3.17 Convenient principal bundles; 3.18 Convenient Lie algebroids; 3.19 Classical results on Banach structures which are not true in the convenient setting; 3.20 Notes 4. Projective limits; 4.1 Projective limits in categories; 4.2 Projective limits of topological spaces; 4.3 Projective limits of Banach and normed spaces; 4.4 Projective limits and linear functionals; 4.5 Projective limits of differential maps; 4.6 Projective limits of Banach manifolds; 4.7 Projective limits of Banach-Lie groups; 4.8 Projective limits of Banach and normed vector bundles; 4.9 The infinite jet bundle; 4.10 Projective limits of Banach principal bundles; 4.11 The Fr\'{e}chet space $\mathcal {H}( \mathbb{F}_1, \mathbb{F}_2) $ and the Banach space $\mathcal {H}_b(\mathbb{F}_1, \mathbb{F}_2)$; 4.12 Projective limits of generalized frame bundles; 4.13 Projective limits of $G$-structures; 4.14 Projective limits of tensor structures; 4.15 Darboux charts on a projective limit; 4.16 Examples and counter example; 4.17 Projective limits of Finsler-Banach manifolds; 4.18 Projective limits of anchored bundles and Lie algebroids; 4.19 Notes 5. Direct limits; 5.1 Direct limits of categories; 5.2 Direct limits of topological spaces; 5.3 Ascending sequences of normed spaces; 5.4 Ascending sequences of Banach spaces; 5.5 Differential equations on direct limits of Banach spaces; 5.6 Direct limits of ascending sequences of Banach manifolds; 5.7 Direct limits of ascending sequences of topological groups; 5.8 The Fr\'{e}chet topological group $\operatorname {G}(\mathbb{E})$; 5.9 Direct limits of ascending sequences of Banach-Lie groups; 5.10 Lie subgroups of $\operatorname{G}(\mathbb{E})$; 5.11 Direct limits of Banach and normed vector bundles; 5.12 Direct limits of Banach connections; 5.13 Direct limits of Banach principal bundles; 5.14 Direct limits of frame bundles; 5.15 Direct limits of $G$-structures; 5.16 Direct limits of tensor structures; 5.17 Examples of direct limits of tensor structures <br />5.18 Symplectic forms on direct limits of ascending sequences; 5.19 Direct limits of anchored bundles and Lie algebroids; 5.20 Notes 6. Convenient Lie algebroids and prolongations; 6.1 $\mathcal{A}$-connections on a convenient bundle; 6.2 Foliations and Banach-Lie algebroids; 6.3 Pseudo-Riemannian and Riemannian structures on a Banach-Lie algebroid; 6.4 Prolongation of a convenient Lie algebroid along a fibration; 6.5 Projective limits of prolongations of Banach Lie algebroids; 6.6 Direct limits of prolongations of Banach Lie algebroids; 6.7 Notes 7. Partial Poisson structures; 7.1 Partial Poisson manifolds; 7.2 Partial Lie algebroids and partial Poisson manifolds; 7.3 Cohomology associated to a partial Poisson manifold; 7.4 Convenient Poisson morphisms; 7.5 Projective limits of partial Poisson Banach manifolds; 7.6 Direct limits of partial Poisson Banach manifolds; 7.7 Notes 8. Integrability of distributions; 8.1 On the problem of integrability of a distribution; 8.2 Integrability and invariance; 8.3 Integrability and Lie invariance; 8.4 Applications to Banach-Lie algebroids and Poisson manifolds; 8.5 Criterion of integrability on submersive projective limits; 8.6 Criterion of integrability on direct limits; 8.7 Almost symplectic foliation on direct limit 8.8 Notes A. Fr\'{e}chet spaces; B. Categories; C. Sheaves theory; D. Locally convex vector bundles <br />E. The KdV equation; F. Formal integrability of systems of PDEs Further studies Bibliography Index
• Publisher Details: Boca Raton : CRC Press, Taylor & Francis Group
• Publication Date: 2024
• Extent: 1 online resource, illustrations
• Subjects: 512.55; Banach algebras; MATHEMATICS / Geometry / General; MATHEMATICS / Geometry / Differential; MATHEMATICS / Mathematical Analysis
• Languages: English
• ISBNs: 9781003435587; 1003435580; 9781000965988; 1000965988; 9781000966015; 1000966011
• Related ISBNs: 9781032561714; 9781032564500
• Notes: Note: Includes bibliographical references and index.; Note: Vendor-supplied metadata.
• Access Rights: Legal Deposit; Only available on premises controlled by the deposit library and to one user at any one time; The Legal Deposit Libraries (Non-Print Works) Regulations (UK).
• Access Usage: Restricted: Printing from this resource is governed by The Legal Deposit Libraries (Non-Print Works) Regulations (UK) and UK copyright law currently in force.