Comprehensive Exam Real Analysis January 2000

A passing mark will be given if 5 problems are completely solved.

1. If X is a Banach space and x is any element of X then there is a functional x^{∗} ∈ X^{∗} such that ||x^{∗}|| = 1 and x^{∗}(x) = ||x||. Show that this happens in both the Banach space C[0, 1] and in the Banach space L^{1}[0, 1]. More specifically do the following:

(a) If f ∈ C[0, 1] describe a functional x^{∗} ∈ C[0, 1]^{∗} so that x^{∗}(f) = ||f||_{∞}.

(b) If f ∈ L^{1}[0, 1] describe a functional x^{∗} ∈ L^{1}[0, 1]^{∗} so that x^{∗}(f) = ||f||_{1}.

2. Let (A_{n}) be a sequence of sets in an algebra A of sets. Show that there is a disjoint sequence (B_{n}) of sets in A such that, for each k,

∪_{n=1}^{k}A_{n} = ∪_{n=1}^{k}B_{k}.

3. Consider the following 3 measures, μ_{0}, μ_{1}, and μ_{2}, defined on the measurable space

([0, 1], M) of Lebesgue measureable sets in [0, 1].

μ_{0} is Lebesgue measure.

μ_{1} is the counting measure defined on M. μ_{2} is defined for E ∈ M by μ_{2}(E) = ∫_{E} x^{2} dx.

Indicate for exactly which ordered pairs of integers i and j it is true that μ_{i} ≪ μ_{j}.

4. Let (f_{n})_{n=1}^{∞} be a sequence of measurable real-valued functions on [0, 1] such that f_{n} ∈ L_{2}[0, 1] for all n ∈ N and ∑_{n=1}^{∞} ∫ |f_{n}| < ∞.

(a) Show that f = lim_{N→∞} ∑_{n=1}^{N} f_{n} is defined and finite a.e.

(b) Show that f, defined as in the previous part, need not be in L_{2}[0, 1].

5. Let E be a Borel measurable subset of [0, 1] and let χ represent the characteristic or indicator function of E, i.e., χ(x) = 1 if x ∈ E and χ(x) = 0 otherwise. Show that for any ϵ > 0, there is a continuous function f ∈ C[0, 1] such that

∫_{0}^{1} |f − χ|dx < ϵ.

6. Let K = [0, 1] × [0, 1] be the unit square. Let f ∈ C(K) and ϵ > 0 be given. Show that there is a function g ∈ C(K) such that ||f − g||_{∞} < ϵ and g has the the form

g(x, y) = ∑_{i=1}^{n} a_{i}f_{i}(x)g_{i}(y)

where f_{i} and g_{i} are in C[0, 1].