MATH 354
ASSIGNMENT 2, SOLUTIONS
Problems
Please justify carefully your answers.
1. [10 points] Is it true that in any metric space (X, d),
cl(D(x, r) = cfw_y X : d(x, y ) r.
Provide a proof or nd a counterexample.
Solution. Counterexample. Let X be a set
MATH 354
SOLUTIONS TO ASSIGNMENT 3
1. [10 points] Let X be the collection of all sequences of positive integers. If x = (nj )
j =1
and y = (mj ) are two elements of X , set
j =1
k (x, y ) = inf cfw_ j : nj = mj
and
0
d(x, y ) =
1
k(x,y )
if x = y
if x =
(Six) is EX~2§ s ix~yi+ly~zl s IX~yE+5iyl+e
So Mr) may) é ix A 3/} -§+~ E for any 5 > 0, so 5(x) ~ (My) g «1/. Symmet-
rically, we obtain that L50) - My}! \<\ {x «3/1, and (5 satisfies the described
Lipschitz condition, so 5 is continuous. [:1
(b)
McGill University
Math 354: Honors Analysis 3
An estimate for the Intermediate Value theorem
We shall prove an estimate (Drurys Math 354 notes, theorem 137) that is used to prove the
Intermediate Value theorem in normed vector spaces.
Theorem. Let X be a
Math 354, Honors Analysis, D. Jakobson
Short summary: things to review and remember
A distance (p, q) between points p, q in a metric space satises
.
dist(p, q) > 0 if p = q; dist(p, p) = 0.
dist(p, q) = dist(q, p).
dist(p, q) + dist(q, r) dist(p, r).
McGill University
Math 354: Honors Calculus 3
Various things about metric spaces
I. Application of Stone-Weierstrass theorem.
Let T = R(mod2) be the unit circle (or 1-dimensional torus). Consider the set C(T) of continuous functions on T (they may be iden
McGill University
Math 354: Honors Real Analysis 3
Denition. An algebra A of functions is called separating if for any x = y X, there exists f A
such that f (x) = f (y). Suppose A is unital (contains the constant function 1) and separating.
Given two numb
HW 3 DUE OCTOBER 4
A non-negative measurable function f 0 is said to be integrable if,
Z
sup
g(x)dx < .
0gf
R
where the supremum is taken over all measurable g with 0 g f ,
with g in addition bounded and supported on a set of finite measure.
For an integr
HW 2
HONORS ANALYSIS 3
DUE TUESDAY 27TH SEPTEMBER 2016
(1) A G set is a countable intersection of open sets. Countable
unions of closed sets are known as F sets. Prove that a closed
set is G set and an open set an F set. Show that there exists
F sets whic
HONORS ANALYSIS 3
ASSIGNMENT 1
DUE SEPT. 20 IN CLASS
(1) If E is measurable, and > 0 is given, define
E = cfw_x : x E
Prove that E is measurable and that (E) = (E).
(2) Suppose that E is a measurable set, and
1
On = cfw_x : d(x, E) <
n
where d(x, E) = in
MATH 354, Assignment 4, Solutions
October 26, 2013
1. Let x X , and consider the iterates f n (x). Since f is an isometry, f m is also an isometry for any
m N. Then given m, n N, n > m,
d(f n (x), f m (x) = d(f m (f nm (x), f m (x) = d(f nm (x), x).
Dene
MATH 354
ASSIGNMENT 1, short solutions
1. [10 points] Let X be the collection of all sequences of positive integers. If x = (nj )
j =1
and y = (mj ) are two elements of X , set
j =1
k (x, y ) = inf cfw_ j : nj = mj
and
d(x, y ) =
0
1
k(x,y )
if x = y
if
McGill University
Math 354: Honors Analysis 3
Practice problems
not for credit
Problem 1. Determine whether the family of F = cfw_fn functions fn (x) = xn is uniformly equicontinuous.
1st Solution: The family F is clearly uniformyl bounded. If it were un
McGill University
Math 354: Honors Analysis 3
Assignment 4
Fall 2012
due Friday, October 19
Problem 1 (extra credit). Let X = C 1 [0, 1] denote the space of continuously dierentiable
functions on [0, 1].
a) Prove that the expression
|f |2 = max |f (x)| +
McGill University
Math 354: Honors Analysis 3
Assignment 3
Fall 2012
Solutions to selected problems
Problem 1. Lipschitz functions. Let LipK be the set of all functions continuous functions on
[0, 1] satisfying a Lipschitz condition with constant K > 0, i
McGill University
Math 354: Honors Analysis 3
Assignment 2
Fall 2012
Solutions to selected problems
Problem 1. Let K be the Cantor set. Prove that
a) Prove that the endpoints of the intervals that appear in the construction of K , i.e. the points
0, 1, 1/
McGill University
Math 354: Honors Analysis 3
Assignment 1
Fall 2012
Solutions to selected Exercises
Exercise 1.
(i) Verify the identity for any two sets of complex numbers cfw_a1 , ., an and cfw_b1 , ., bn
2
n
ak bk
n
n
a2
k
=
k=1
b2
k
k=1
k=1
1
2
n
n
McGill University
Math 354: Honors Analysis 3
Dierentiation in Banach Spaces:
Example
Problem 1 Let : C ([0, 1]) C ([0, 1]) be given by (f ) = f 3 . Prove that is dierentiable, and
compute D.
Solution: Let f, h C ([0, 1]). D(f ) (if it exists) applied to
SPECIAL SETS & SPACES & THEIR PROPERTIES
ROB
Abstract. Here we wish to provide an extensive list of general topological/metric spaces X , stating some
properties and special cases of these spaces relevant to an intermediate level real analysis course. The
McGill University
Math 354: Honors Calculus 3
Bernstein Approximation Theorem. (S. Drurys Math 354 notes, Theorem 86). Let f (x)
C ([0, 1]). Let the n-th Bernstein approximation polynomial Bn (f, x) be dened by
n
n
f
k
Bn (f, x) =
k=0
k
n
xk (1 x)nk .
Th
McGill University
Math 354: Honors Analysis 3
Baires Category Theorem and Uniform Boundedness Principle
I. Baires Category Theorem.
Theorem 1 (Baires Category Theorem). (Drury, Theorem 61). Let X be a complete metric
space, and let Ak be a closed subset o
McGill University
Math 354: Honors Analysis 3
Inverse Function theorem in Rn .
Our exposition follows that in Rudins book.
Theorem 1 (Inverse Function Theorem). Let Rn be an open set, and
let F : Rn be continuously dierentiable. Let a , let b = F (a), and
Pr. Dmitry Jakohson Leo Raymond-Belzile, 260424.542
Honours Analysis 3 MATH 354 Assignment 5: due November 137th
2. The Cantor set C can also he described in terms of ternary expansions.
(h) The Cantor-Lebesgue function is defined on C by
.30 CC