Advanced Calculus 1: homework # 7This homework will not be graded. It is just for your practice79. Prove that the trigonometric polynomialsT (x) =nXk=0akcos kx +nXk=0bksin kx, ak, bk∈ IRform an algebra. Hint: cos x + i sin x = eix.80. Let S1= {z ∈ C : |z| = 1} be the unit circle in the complex plane. Let A be thealgebra of functions of the formf(eiθ) =NXn=0cneinθ, cn∈ C, θ ∈ IR.It is easy to see that f ≡ 1 belongs of A and A separates points (do not prove it). Provethat there are complex valued functions on S1that cannot be uniformly approximatedby functions in A. Hint: For f ∈ AZ2π0f(eiθ)eiθdθ = 0 .81. Prove that complex polynomialsp(z) =NXn=0cnzn, cn∈ Care not dense in C(D, C), whereD = {z ∈ C : |z| ≤ 1}is the unit disc in C. Hint: Consider f(z) = z. Is the previous exercise helpful?82. We know that if f : [a, b] → IR is continuous andZbaf(x)xndx = 0 (1)for n = 0, 1, 2, 3, . . ., then f(x) = 0 for all a ≤ x ≤ b. We proved it using the Weierstrasstheorem. Suppose now that f : [a, b] → IR is continuous and (1) holds for all n ≥ 2011.Does it follow that f(x) = 0 for all a ≤ x ≤ b?83. Prove that if f : [0, 1] → IR is such thatZ10f(x)enxdx = 0 for all n = 0, 1, 2, . . .,then f(x) = 0 for all 0 ≤ x ≤ 1. Provide two proofs following the methods:(a) Use the Stone-Weierstrass theorem.(b) Use the change of variables formula and apply the Weierstrass theorem.84. Show that there is a unique continuous real valued function f : [0, 1] → IR suchthatf(x) = sin x +Z10f(y)ex+y+1dy.85. Let (X, d) be a nonempty complete metric space. Let S : X → X be a givenmapping and write S2for S ◦S i.e. S2(x) = S(S(x)). Suppose that S2is a contraction.Show that S has a unique fixed point.86. Let E be a compact set and let F ⊂ C(E, IR) be an equicontinuous family offunctions. Does it imply that the family F is bounded in C(E, IR)?87. Let f : IRn→ IR be bounded and uniformly continuous. Prove that the family offunctions {gz}z∈IRn, gz(x) = f (x)f (x − z) is equicontinuous.88. Suppose E is a compact metric space and fn: E → IR, n = 1, 2, . . . is a boundedand equicontinuous sequence of functions. Suppose that fnconverges pointwise to acontinuous function f : E → IR (i.e. fn(x) → f(x) for every x ∈ E). Prove directly(i.e. without using Arzela-Ascoli theorem) that fn⇒ f uniformly on E.89. Suppose fn: IR → IR, n = 1, 2, . . . is a bounded and equicontinuous sequence offunctions. Suppose that fnconverges pointwise to a continuous function f : IR → IR.Does it imply that fn⇒ f uniformly on IR?90. Let fn: [a, b] → IR be a sequence of increasing functions that is pointwise con-vergent to a continuous function f : [a, b] → IR. Prove that fn⇒ f uniformly on[a, b].91. Let {fn} be a sequence of real valued C1functions on [0, 1] such that, for all n,|f0n(x)| ≤1√x(0 < x ≤ 1),Z10fn(x) dx = 0.Prove that the sequence has a subsequence that converges uniformly on [0,