Gx Sx N. Restriction of a convex function to a line f Rn → R is convex if and only if the function g R → R, g(t) = f(xtv), domg = {t xtv ∈ domf} is convex (in t) for any x ∈ domf, v ∈ Rn can check convexity of f by checking convexity of functions of one variable. X Sn Y ⇐⇒ Y −X positive semidefinite these two types are so common that we drop the subscript in K properties many properties of K are similar to ≤ on R, eg, x K y, u K v =⇒ xu K y v Convex sets 2–17 Minimum and minimal elements K is not in general a linearordering we can have x 6 K y and y 6 K x.
That is, f(x 1) = f(x 2) implies x 1 = x 2 In other words, every element of the function's codomain is the image of at most one element of its domain The term onetoone function must not be confused with onetoone correspondence that. X Sn Y ⇐⇒ Y −X positive semidefinite these two types are so common that we drop the subscript in K properties many properties of K are similar to ≤ on R, eg, x K y, u K v =⇒ xu K y v Convex sets 2–17 Minimum and minimal elements K is not in general a linearordering we can have x 6 K y and y 6 K x. If X = PN i=1 Xi, N is a random variable independent of Xi’s Xi’s have common mean µ Then EX = ENµ • Example Suppose that the expected number of accidents per week at an industrial plant is four Suppose also that the numbers of workers injured in each accident are independent random variables with a common mean of 2.
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Definition A sequence of functions fn X → Y converges uniformly if for every ϵ > 0 there is an Nϵ ∈ N such that for all n ≥ Nϵ and all x ∈ X one has d(fn(x), f(x)) < ϵ Uniform convergence implies pointwise convergence, but not the other way around For example, the sequence fn(x) = xn from the previous example converges pointwise. Sequences of Functions Uniform convergence 91 Assume that f n → f uniformly on S and that each f n is bounded on S Prove that {f n} is uniformly bounded on S Proof Since f n → f uniformly on S, then given ε = 1, there exists a positive integer n 0 such that as n ≥ n 0, we have f n (x)−f (x) ≤ 1 for all x ∈ S (*) Hence, f (x) is bounded on S by the following. 1 ｽ ｽ ｽI ｽ ｽ ｽ 57 T1600 ｽc ｽd 1337 03 ｽb ｽi ｽV ｽ ｽ ｽR ｽE ｽG ｽh ｽ ｽ ｽ ｽh ｽj 1011. If X = PN i=1 Xi, N is a random variable independent of Xi’s Xi’s have common mean µ Then EX = ENµ • Example Suppose that the expected number of accidents per week at an industrial plant is four Suppose also that the numbers of workers injured in each accident are independent random variables with a common mean of 2.
