Direct and Projective Limits of Geometric Banach Structures. (Chapman & Hall/CRC Monographs and Research Notes in Mathematics)

Cover
Half Title
Series Page
Title Page
Copyright Page
Dedication
Contents
Preface
Authors
Acknowledgements
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
	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
	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 Fréchet Space H(F1, F2) and the Banach Space Hb (F1, F2)
	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. Direct Limits of Ascending Sequences of Banach-Lie Groups
	5.9. The Fréchet Topological Group G(E)
	5.10. Lie Subgroups of G(E)
	5.11. Direct Limits of Banach and Normed Vecto 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
	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. 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 with 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, 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 a Direct Limit
	8.8. Notes
A. Fréchet Spaces
	A.1. The Topology of Fréchet Spaces
	A.2. Differential Equations on Fréchet Spaces
B. Categories
C. Sheaves Theory
	C.1. Presheaves
	C.2. Sheaves
	C.3. Projective and Direct Limits of Sheaves
D. Locally Convex Vector Bundles
E. The KdV Equation
F. Formal Integrability of Systems of PDEs
Further Studies
Bibliography
Index