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Introduction to Real Analysis, Fall 2014. Midterm #2.
Due Friday, November 14th by 4pm
Directions: Provide complete arguments (do not skip steps). State
clearly and FULLY any result you are referring to. Partial credit for incorrect solutions, containing steps in the right direction, may be given. If you
are unable to solve a problem (or a part of a problem), you may still use its
result to solve a later part of the same problem or a later problem in the
exam.
Scoring system: Exam consists of 4 problems, worth 12,12,16 and 12
points. If s1 , s2 , s3 , s4 are your individual scores in decreasing order, your
total is s1 + s2 + s3 + s4 /2. Thus, the maximal possible total is 46, but the
score of 40 counts as 100%.
Rules: You are NOT allowed to discuss midterm problems with anyone
else except me. You may ask me any questions about the problems (e.g.
if the formulation is unclear), but I may only provide minor hints. You
may freely use your class notes, previous homework assignments (including
posted solutions) and Rudin’s book. The use of other books or any online
sources (besides the course webpage) is prohibited.
Hint: Go over previous homeworks before working on exam problems.
Some problems are very relevant.
1. (12 pts) Let (X, d) be a metric space. Recall that given a non-empty
subset Y of X and a point a ∈ X, the distance d(a, Y ) from a to Y is defined
by d(a, Y ) = inf{d(a, y) : y ∈ Y }.
(a) (6 pts) Suppose that closed balls in X are compact (recall that a
closed ball is a set of the form {x ∈ X : d(a, x) ≤ M } for some fixed
a ∈ X and M ∈ R). Prove that for any non-empty closed subset Y of
X and for any a ∈ X, there exists y ∈ Y such that d(a, y) = d(a, Y ).
(b) (6 pts) This part shows that the assertion of (a) may be false without
the assumption that closed balls are compact. Let X = C[0, 1], the
space of continuous functions from [0, 1] to R, and d the uniform
metric on X, that is,
d(f, g) = max |f (x) − g(x)|.
x∈[0,1]
n
Let Y = {fn }n∈N where fn (x) = n+1
n x . Prove that Y is closed in X
and d(0, Y ) < d(0, fn ) for any n (where 0 denotes the zero function).
1
2
2. (12 pts) Let a ≤ b be real numbers, let S be a subset of [a, b] which
contains both a and b, and let f : S → R be an increasing function (note: f
is only defined on S).
(a) (4 pts) Prove that f can be extended to an increasing function from
[a, b] to R, that is, there exists an increasing function F : [a, b] → R
such that F (x) = f (x) for all x ∈ S. Hint: Define F by an explicit
formula involving the supremum of values of f on a suitable subset.
(b) (4 pts) Now assume that f (S) is a dense subset of [f (a), f (b)]. Prove
that there exists unique function F satisfying the conclusion of (a).
Then prove that this F is continuous. Hint: Prove uniqueness by
contradiction.
(c) (4 pts) Before doing this problem, read about Cantor set (Rudin,
page 41). Let a = 0, b = 1, let C ⊂ [0, 1] be the Cantor set and let
S = [0, 1] \ C, that is,
S = ( 13 , 23 ) ∪ ( 19 , 29 ) ∪ ( 79 , 89 ) ∪ . . .
Use (b) to prove that there exists an increasing continuous function
F : [0, 1] → [0, 1] such that F (0) = 0, F (1) = 1 and F (S) is the set
of all rational numbers in (0, 1) of the form 2nk (where n, k ∈ Z≥0 )
and draw the graph of such a function.
3. (16 pts) Let (X, d) be a metric space and {fn } a sequence of functions
from X to R. Let S = { n1 : n ∈ N}, let Y = X × S, and define the metric
D on Y by
1
D((a, n1 ), (b, m
)) = max{d(a, b), | n1 −
1
m |}.
Define the function F : Y → R by F ((x, n1 )) = fn (x).
(a) (2 pts) Prove that D is indeed a metric on Y .
(b) (2 pts) Prove that for any point (x, n1 ) ∈ Y there exists δ > 0 such
that NδY ((x, n1 )) = NδX (x) × { n1 }.
(c) (2 pts) Prove that for any N ∈ N there exists δ > 0 with the following
1
property: if we are given any two points (x, n1 ), (z, m
) ∈ Y with
1
1
D((x, n ), (z, m )) < δ, then n = m or n, m ≥ N .
(d) (3 pts) Use (b) to prove that F is continuous ⇐⇒ each fn is
continuous
(e) (5 pts) Use (c) to prove that F is uniformly continuous ⇐⇒ the
sequence {fn } is equicontinuous and converges uniformly.
(f) (2 pts) Change the definition of metric D on Y so that the following
becomes true: F is uniformly continuous ⇐⇒ {fn } is equicontinuous. An answer (new formula for D) is sufficient.
4. (12 pts) The goal of this problem is to deduce the “strong” StoneWeierstrass Theorem from the “weak” Stone-Weierstrass Theorem as formulated below.
Theorem (Stone-Weierstrass): Let X be a compact metric space and let
A ⊆ C(X) be an algebra.
(i) (weak version) Assume that A separates points on X and contains
(all) constant functions. Then for any f ∈ C(X) and ε > 0 there is
g ∈ A such that d∞ (f, g) < ε.
(ii) (strong version) Assume that A separates points on X and vanishes
nowhere on X. Then for any f ∈ C(X) and ε > 0 there is g ∈ A
such that d∞ (f, g) < ε.
So, assume that A satisfies the hypothesis of (ii).
(a) (5 pts) Prove that A contains a function H such that H(x) 6= 0 for
all x ∈ X. Hint: Use nowhere vanishing of A, compactness of X
and the fact that the sum of squares of real numbers y12 + . . . + yn2 is
zero only if each yi = 0.
(b) (2 pts) Let A0 be the set of all functions of the form f + c with f ∈ A
and c ∈ R (thought of as a constant function). Prove that A0 is an
algebra which contains constant functions.
(c) (5 pts) Now prove the strong version of the Stone-Weiestrass theorem. Hint: Given f ∈ C(X) and ε > 0, to find g ∈ A with
d∞ (f, g) < ε, start by applying the weak Stone-Weiestrass theorem
to the algebra A0 and suitable function involving f and H (where H
is the function from part (a)).