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Mat 201b Recap

4. Let (Ω,Σ, μ) be a measure space. If f : Ω → R, then we say that f is a measurable function if for every t ∈ R, the level set: Sf (t) = {x ∈ Ω|f(x) > t} is measurable. f : Ω→ C is measurable if and only if both Re(f) and Im(f) are measurable. 5. A real function f : Ω → R is lower semicontinuous on Ω if Sf (t) is open, and is upper semicontinuous if {x ∈ Ω|f(x) < t} is open. Equivalently, f is...

Mat 215a Recap

(a) D: α0 = 1, α1 = 1, α2 = 1. χ(D ) = 1. (b) S: α0 = 1, α1 = 0, α2 = 1. χ(S ) = 2. (c) Möbius band: α0 = 1, α1 = 2, α2 = 1. χ(M) = 0. (d) Mg of genus g: α0 = 1, α1 = 2g, α2 = 1. χ(Mg) = 2− 2g. 6. RP n = (R − {0})/{z ∼ λz} = S/{z ∼ −z} = S/Z2 RP n = σ ∪ RP n−1 implies RP n = σ ∪ σn−1 ∪ · · · ∪ σ χ(RP ) = 1 for n even and 0 for n odd. 7. CP n = (C − {0})/{z ∼ λz} = S/{z ∼ λz||λ| = 1} CP n = σ ∪ ...

Mat 300 Recitations – Week

MAT 544 Real

whenever ‖~v‖V < δ. In this case the bounded linear operator dF~v0 is called the (Frechet) derivative of F at ~v0. Prove that if ‖ • ‖V is a norm on V which is equivalent to ‖ • ‖V , then F is differentiable at ~v9 with respect to ‖ • ‖V if and only if it is differentiable at ~v0 with respect to ‖ • ‖V , and in this case the derivatives are equal. Similarly, if ‖ • ‖W is a norm on W which is eq...

MAT 145 : Homework Solutions

Problems from Review of Set Theory Notes 3 (c) We prove that A× (B −C) = (A×B)− (A×C). We show that A× (B −C) ⊂ (A×B)−(A×C), and (A×B)−(A×C) ⊂ A×(B−C). For the first containment, let x ∈ A× (B − C). Then x = (a, b) where a ∈ A and b ∈ B − C. We certainly have x ∈ A × B, and since the second coordinate of x is not in C, we conclude that x / ∈ A× C. For the second containment, let x ∈ (A× B)− (A×...