Analysis theorems

Let X be a complete metric space. If ƒ : X → X is a contraction, then ƒ has a unique fixed point.

Let A ∈ R^n×m and b ∈ R^m×k. Then AB ∈ R^n×k and ‖AB‖ ≤ ‖A‖ ‖B‖

Let U open in Rⁿ, V open in R^m with F : U → V and G : V → R^k. If F is differentiable at u ∈ U and F is differentiable at F(u) = v ∈ V, then G ∘ F is differentiable at u. Furthermore if T : Rⁿ → R^m is the first order affine approximation to F at u and S : R^m → R^k is the first order affine approximation to G at v then S ∘ T is the affine function which approximates G ∘ F to first order at u.

Let U be an open subset of Rⁿ and ƒ : U → R differentiable. If the line segment J from a to b is contained in U, then there exists c ∈ J with ƒ(b) − ƒ(a) = ∇ƒ(c) • (b − a).

Let U be open in Rⁿ and F ∈ C¹(U, Rⁿ). If F is smooth, then F is an open function.

Let F ∈ C¹(W, Rⁿ) where W is open in Rⁿ. If p ∈ W is a smooth point of F, then there exists open U ∋ p with U ⊆ W and an open set V such that F : U → V is a smooth homeomorphism. Also F^-1 ∈ C¹(V, U) and D[F^-1](F(a)) = [DF(a)]^-1 for any a ∈ U.

• For a function F ∈ C¹(Ω, R^m) (Ω open in R^n+m) and a ∈ Rⁿ and b ∈ R^m with F(a, b) = 0. If D₂F(a, b) has rank m, then there exists open U ∋ a in Rⁿ and open V ∋ (a, b) in R^n+m and a function F ∈ C¹(U, R^m) such that
• {(x, y) ∈ V : F(x, y) = 0} = {(x, G(x)) : x ∈ U}.
• In addition, DG(x) = -[D₂F(x, G(x))]^-1D₁F(x, G(x))

• Let ƒ ∈ C^m+1(U, R) where U is open in Rⁿ. If the line segment bx ⊂ U, then
• ƒ(x) = p_m(x) + 1/(m + 1)! D^m+1ƒ(c)(u)^m+1
• where u = x − b and c ∈ bx.

• Let F ∈ C¹(Ω, R^m) where Ω is open in R^n+m and S = {x ∈ Ω : F(x) = 0}. For a differentiable function φ : Ω → R, if s ∈ S is a local extreme of φ|_S, the either DF(s) has rank less than m, or there exists a nonzero vector v ∈ R^m (written horizontally) such that
• ∇φ(s) = vDF(s)

Let I be a closed interval in Rⁿ, A ⊂ I with volA = 0, and ƒ : I → R is a bounded function. If ƒ has the property that for any closed interval J ⊆ I, ƒ|_J is integrable whenever J ∩ A = ∅, then ƒ is integrable, ƒ|_I−A is integrable, and ∫_I ƒ = ∫_I−A ƒ.

Let N and Q be closed intervals of Rⁿ and R^m respectively, and let I = N × Q. Let ƒ : I → R be integrable. If for each y ∈ Q, the function g_y : N → R defined by g_y(x) = ƒ(x, y) is integrable, then the function h : Q → R defined by h(y) = ∫_N g_y is integrable and ∫_Q h = ∫_I ƒ.

• Consider a compact A ⊂ U, an open set in Rⁿ, and F ∈ C¹(U, Rⁿ), a smooth one to one function. For any ε > 0, there exists a δ > 0 such that if C is an n-cube with diamC < δ, C ∩ A ≠ ∅ and a ∈ C, then C ⊂ U and
• abs[volF(C)/volC − |J_F(a)|] < ε
• where J_F(a) = detDF(a) is the Jacobian.

• Let U be an open set in Rⁿ and Φ ∈ C¹(U, Rⁿ), a smooth one to one function. If D ⊂ U is a compact Jordan domain and ƒ : φ(D) → R is continuous, then
• ∫_Φ(D) ƒ = ∫_D ƒ ∘ Φ |J_Φ|
• where J_Φ = detDΦ is the Jacobian.