Math 2.0

Global analysis, analysis on manifolds Theorem

Implicit function theorem in Banach spaces

Builds on Inverse function theorem in Banach spacesBanach spaceFréchet derivativeImplicit function theorem

Statement

Let 𝕂\mathbb{K} be ℝ\R or ℂ\C, and let XX, YY and ZZ be Banach spaces over 𝕂\mathbb{K}. Let Ω⊆X×Y\Omega \subseteq X \times Y be open, and let F ⁣:Ω→ZF \colon \Omega \to Z be of class C1C^1 in the sense of the Fréchet derivative. Let DxFD_x F and DyFD_y F be the partial derivatives of FF, with values in the continuous linear maps X→ZX \to Z and Y→ZY \to Z. Let (a,b)∈Ω(a, b) \in \Omega with F(a,b)=0F(a, b) = 0, and suppose that DyF(a,b) ⁣:Y→ZD_y F(a, b) \colon Y \to Z is bijective. Then there are open sets U⊆XU \subseteq X and V⊆YV \subseteq Y with a∈Ua \in U, b∈Vb \in V and U×V⊆ΩU \times V \subseteq \Omega, and a unique map g ⁣:U→Vg \colon U \to V, such that the following holds. For (x,y)∈U×V(x, y) \in U \times V, F(x,y)=0F(x, y) = 0 if and only if y=g(x)y = g(x). After shrinking UU if needed, gg is of class C1C^1, DyF(x,g(x))D_y F(x, g(x)) is bijective for each x∈Ux \in U, and Dg(x)=−DyF(x,g(x))−1 DxF(x,g(x)).Dg(x) = -D_y F(x, g(x))^{-1}\, D_x F(x, g(x)).

Remarks

Apply the inverse function theorem in Banach spaces to Φ(x,y)=(x,F(x,y))\Phi(x, y) = (x, F(x, y)) from Ω\Omega to X×ZX \times Z. Its derivative at (a,b)(a, b) is bijective because DyF(a,b)D_y F(a, b) is, and the local inverse of Φ\Phi has the form (x,z)↦(x,h(x,z))(x, z) \mapsto (x, h(x, z)); then g(x)=h(x,0)g(x) = h(x, 0). The formula for DgDg comes from differentiating F(x,g(x))=0F(x, g(x)) = 0 with the chain rule.

Example: let h∈C([0,1])h \in C([0,1]) and F(λ,u)=u−h−λsin⁡∘uF(\lambda, u) = u - h - \lambda \sin \circ u on ℝ×C([0,1])\R \times C([0,1]). Then F(0,h)=0F(0, h) = 0 and DuF(0,h)=id⁡D_u F(0, h) = \id. So for λ\lambda near 00, the equation u=h+λsin⁡(u)u = h + \lambda \sin(u) has exactly one solution u=g(λ)u = g(\lambda) near hh, and gg is C1C^1.

For finite-dimensional XX, YY and ZZ, the theorem is the implicit function theorem of several-variable calculus.

Going deeper

For a student

The theorem is the basic tool of perturbation theory. A solution of a problem at one parameter value persists for nearby parameters when the linearized problem is uniquely solvable. The solution then depends smoothly on the parameter. Bifurcation theory studies exactly the points where DyFD_y F fails to be invertible. It is equivalent to the inverse function theorem in Banach spaces, which is the special case F(x,y)=f(y)−xF(x, y) = f(y) - x.

Reading path

To understand this page from the ground up, read these 54 pages in order. The foundations come first.

  1. Banach space concept
  2. Bijective function concept
  3. Continuous linear map concept
  4. Element definition
  5. Euclidean space object
  6. Extended real number line object
  7. Function concept
  8. Function composition concept
  9. Inverse function concept
  10. Operator norm concept
  11. Proposition concept
  12. Real numbers object
  13. Set bounded above definition
  14. Bounded inverse theorem statement
  15. Set definition
  16. Fréchet derivative definition
  17. Total derivative concept
  18. Function of several variables concept
  19. Implication definition
  20. Predicate definition
  21. Supremum definition
  22. Ordered pair definition
  23. Fréchet derivative concept
  24. Limit of a function of several variables concept
  25. Partial derivative concept
  26. Existential quantifier definition
  27. Universal quantifier definition
  28. Cartesian product definition
  29. Differentiable function of several variables definition
  30. Partial derivative definition
  31. Subset definition
  32. Uniqueness quantifier definition
  33. Differentiable on an open set definition
  34. Binary relation definition
  35. Higher partial derivatives definition
  36. Jacobian matrix definition
  37. Chain rule in several variables statement
  38. Iterated derivative definition
  39. Function definition
  40. Jacobian determinant definition
  41. Derivative of the inverse function statement
  42. C^k function characterised by continuous derivatives statement
  43. Metric space definition
  44. Convergence of a sequence in a metric space definition
  45. Lipschitz map definition
  46. Inverse function theorem statement
  47. Diameter of a set definition
  48. Contraction mapping definition
  49. Implicit function theorem statement
  50. Epsilon characterisation of Cauchy sequences statement
  51. Complete metric space definition
  52. Banach fixed point theorem statement
  53. Inverse function theorem in Banach spaces statement
  54. Implicit function theorem in Banach spaces statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Implicit_function_theorem
About this page
TagBTM6WJ0A52
URIm2://BTM6WJ0A52
Kindstatement, theorem
WritingW0 stub
MSC58C15, 47J07, 46G05
StatusThis copy of the site has no build data. Run m2 status BTM6WJ0A52.
AboutFréchet derivative, Banach space
UsesInverse function theorem in Banach spaces, Banach space, Fréchet derivative, Implicit function theorem
Changed2 November 2026

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