Math 2.0

Global analysis, analysis on manifolds Definition

Banach manifold

Builds on Banach spaceFréchet derivativeHomeomorphismOpen setHausdorff spaceSmooth manifold

Definition

Let MM be a Hausdorff space and let k≥1k \ge 1 be an integer or ∞\infty. A chart on MM is a pair (U,φ)(U, \varphi) with U⊆MU \subseteq M open. Here φ\varphi is a homeomorphism from UU onto an open subset of a real Banach space EE. A CkC^k atlas on MM is a family of charts (Ui,φi)i∈I(U_i, \varphi_i)_{i \in I} such that:

  1. The sets UiU_i cover MM.
  2. For all i,j∈Ii, j \in I, the transition map φj∘φi−1 ⁣:φi(Ui∩Uj)→φj(Ui∩Uj)\varphi_j \circ \varphi_i^{-1} \colon \varphi_i(U_i \cap U_j) \to \varphi_j(U_i \cap U_j) is of class CkC^k in the sense of the Fréchet derivative.

Two CkC^k atlases are equivalent if their union is a CkC^k atlas. A Banach manifold of class CkC^k is a space MM with an equivalence class of CkC^k atlases. If all charts take values in the same Banach space EE, the manifold is modelled on EE.

Remarks

When every model space is ℝn\R^n, this is the definition of a smooth manifold of class CkC^k. When every model space is a Hilbert space, MM is a Hilbert manifold. Some authors start from a set instead of a Hausdorff space and give it the topology defined by the charts.

Example: an open subset UU of a Banach space EE, with the single chart (U,id⁡)(U, \id), is a Banach manifold. So the invertible elements of the Banach algebra of bounded operators on EE form a Banach manifold, since they form an open subset. The unit sphere of a Hilbert space is a Banach manifold, with charts given by projections onto tangent hyperplanes.

Non-example: on ℝ\R, the charts (ℝ,id⁡)(\R, \id) and (ℝ,x↦x3)(\R, x \mapsto x^3) do not form a C1C^1 atlas together. The transition map x↦x1/3x \mapsto x^{1/3} is not differentiable at 00, so clause 2 fails.

Reading path

To understand this page from the ground up, read these 35 pages in order. The foundations come first.

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Banach space, concept, no evidence levelBanach spaceconceptElement, definition, no evidence levelElementdefinitionEuclidean space, object, no evidence levelEuclidean spaceobjectHomeomorphism, concept, no evidence levelHomeomorphismconceptOpen set, concept, no evidence levelOpen setconceptProposition, concept, no evidence levelPropositionconceptReal numbers, object, no evidence levelReal numbersobjectSmooth manifold, concept, no evidence levelSmooth manifoldconceptSet, definition, no evidence levelSetdefinitionTotal derivative, concept, no evidence levelTotal derivativeconceptConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionEmpty set, object, no evidence levelEmpty setobjectFamily of sets, definition, no evidence levelFamily of setsdefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionFréchet derivative, concept, no evidence levelFréchet derivativeconceptIntersection, definition, no evidence levelIntersectiondefinitionUnion, definition, no evidence levelUniondefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUnion of a family of sets, definition, no evidence levelUnion of a family of …definitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionTopological space, definition, no evidence levelTopological spacedefinitionFunction, definition, no evidence levelFunctiondefinitionMetric space, definition, no evidence levelMetric spacedefinitionOpen ball, definition, no evidence levelOpen balldefinitionMetric topology, definition, no evidence levelMetric topologydefinitionOpen set, definition, no evidence levelOpen setdefinitionHausdorff space, definition, no evidence levelHausdorff spacedefinitionBanach manifold, definition, no evidence levelBanach manifolddefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Banach space concept
  2. Element definition
  3. Euclidean space object
  4. Homeomorphism concept
  5. Open set concept
  6. Proposition concept
  7. Real numbers object
  8. Smooth manifold concept
  9. Set definition
  10. Total derivative concept
  11. Conjunction definition
  12. Disjunction definition
  13. Implication definition
  14. Predicate definition
  15. Empty set object
  16. Family of sets definition
  17. Ordered pair definition
  18. Fréchet derivative concept
  19. Intersection definition
  20. Union definition
  21. Existential quantifier definition
  22. Universal quantifier definition
  23. Cartesian product definition
  24. Subset definition
  25. Union of a family of sets definition
  26. Uniqueness quantifier definition
  27. Binary relation definition
  28. Topological space definition
  29. Function definition
  30. Metric space definition
  31. Open ball definition
  32. Metric topology definition
  33. Open set definition
  34. Hausdorff space definition
  35. Banach manifold definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Banach_manifold
About this page
TagVB95VVVDM8
URIm2://VB95VVVDM8
Kinddefinition
WritingW0 stub
MSC58B99, 46T05
DefinesBanach manifold
UsesBanach space, Fréchet derivative, Homeomorphism, Open set, Hausdorff space, Smooth manifold
Changed2 November 2026

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