Analysis theorems

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Analysis theorems
2010-12-14 20:27:35
Math Analysis

Some of the important theorems from Math 342
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  1. Contraction Mapping Theorem
    Let X be a complete metric space. If ƒ : X → X is a contraction, then ƒ has a unique fixed point.
  2. Cauchy-Schwartz for matrices
    Let A ∈ Rn×m and b ∈ Rm×k. Then AB ∈ Rn×k and ‖AB‖ ≤ ‖A‖ ‖B‖
  3. Chain Rule
    Let U open in Rn, V open in Rm with F : U → V and G : V → Rk. 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 → Rm is the first order affine approximation to F at u and S : Rm → Rk 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.
  4. Mean Value Theorem for Real Valued Functions
    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).
  5. Open Mapping Theorem
    Let U be open in Rn and F ∈ C1(U, Rn). If F is smooth, then F is an open function.
  6. Inverse Function Theorem
    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.
  7. Implicit Function Theorem
    • For a function F ∈ C1(Ω, Rm) (Ω open in Rn+m) and a ∈ Rn and b ∈ Rm 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 Rn+m and a function F ∈ C1(U, Rm) 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))
  8. Lagrange Remainder Theorem
    • Let ƒ ∈ Cm+1(U, R) where U is open in Rn. If the line segment bx ⊂ U, then
    • ƒ(x) = pm(x) + 1/(m + 1)! Dm+1ƒ(c)(u)m+1
    • where u = x − b and c ∈ bx.
  9. Lagrange Multipliers Theorem
    • Let F ∈ C1(Ω, Rm) where Ω is open in Rn+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 ∈ Rm (written horizontally) such that
    • ∇φ(s) = vDF(s)
  10. Don’t Sweat the Small Stuff Lemma
    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.
  11. Fubini’s Theorem
    Let N and Q be closed intervals of Rn and Rm respectively, and let I = N × Q. Let ƒ : I → R be integrable. If for each y ∈ Q, the function gy : N → R defined by gy(x) = ƒ(x, y) is integrable, then the function h : Q → R defined by h(y) = ∫N gy is integrable and ∫Q h = ∫I ƒ.
  12. Inner Cube Lemma
    • 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 − |JF(a)|] < ε
    • where JF(a) = detDF(a) is the Jacobian.
  13. Change of Variables
    • 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 |JΦ|
    • where JΦ = detDΦ is the Jacobian.