MATH 521, Spring 2014, Assignment 8Due date: Monday, May 5Submit only the starred (*) questions! (Honors students submitthe Honors Question as well.)1.* Suppose f : [a, b] 7→ R is bounded on [a, b]. Prove that if f is Riemannintegrable on [a, b] then the functionF (x) =Zxaf(t) dtis uniformly continuous on [a, b].2. Suppose f : [0, 2] 7→ R is continuous on [0, 2], and thatR20f(t) dt = 2.Show that there is a c ∈ (0, 2) such that f(c) = c. [Hint: Consider thefunction F (x) =Rx0(f(t) − t) dt and apply the FTC.)3.* Use the definition of Riemann integrability, or an equivalent formula-tion, to prove thatZbax dx =12b2− a2.4. Suppose that f, g : [a, b] 7→ R are Riemann integrable on [a, b]. Definethe following:kfk2=sZbaf(x)2dxhf, gi =Zbaf(x)g(x) dx.(There are the basis for the L2metric space on functions. Note thatthe objects in this space are functions, rather than vectors as theywere in Rn. Nevertheless, we can see immediate parallels with theEuclidean norm k · k2and dot product which is often written in innerproduct notation h·, ·i.)(a) Prove directly that, if f(x) is Riemann integrable on [a, b], thenf(x)2is Riemann integrable on [a, b]. (This guarantees that wemay evaluate kfk2.)(b)* Compute kfnk2, n ∈ N, for the family of functionsfn(x) =1 + nx, for − 1/n < x ≤ 01 − nx, for 0 < x ≤ 1/n0, otherwiseon the interval [−1, 1]. What happens as n → ∞? Does this makesense in the context of the individual functions fn(x)? Are thereare peculiarities? (Note: The norm is a measure of “distance” ofa function from the trivial “zero function” f(x) = 0.)(c)* Prove that |hf, gi| ≤ kfk2· kgk2. Hint: It will help to look backat the proof of the Cauchy-Schwarz Inequality contained in theWeek 4 notes. Consider starting with the identityZbaZba(f(x)g(y) − f(y)g(x))2dy dx ≥ 0.(d) Prove that kf + gk2≤ kfk2+ kgk2. (Hint: use (c)!)Honors! Define the set L2([a, b]) to be the set of all functions which areRiemann integrable on [a, b]. Define d : L2([a, b])×L2([a, b]) 7→ Raccording tod(f, g) = kf − gk2.Show that d(f, g) satisfies all the metric space condition exceptd(f, g) = 0 implies f(x) = g(x). (There are subtleties with thisfinal metric space condition which require some degree of mea-sure theory to