Math 522 – Fall 2014
Homework 3: Due 9/24/2014
1. From Rudin, Chapter 7: #11, 15, 16, 18
2. Let gn (x) = xn . In this problem, you will explore the equicontinuity of
these sequence.
(a) Show that the sequence {gn } is not equicontinuous on [0, 1].
(b) Find a subset of [0, 1] such that {gn } is equicontinuous or prove that
it cannot be equicontinuous.
3. Prove that the set of polynomials in C([a, b], R) is not open. Can a subset
of a metric space ever be both open and dense?
4. Suppose {fn } is a sequence of functions differentiable on [a, b] with continuous derivatives and such that {fn (x0 )} converges for some point x0 ∈
[a, b]. Without using Theorem 7.17 in Rudin, show that if {fn′ } converges
uniformly on [a, b], then {fn } converges uniformly on [a, b] to a function
f and
f (x) = lim fn′ (x) for a ≤ x ≤ b.
n→∞
Honors Problems: Do not turn these in with the rest of your homework.
Hold onto your solutions for now.
1. From Rudin, Chapter 7: #17, 19
2. Let B ⊂ C([0, 1], R) be a closed, bounded, and equicontinuous set. Let
I : B → R be defined by
I(f ) =
Z
1
f dx.
0
Show that there is an f0 ∈ B such that the value of I is maximized.
1