FACULTY OF SCIENCE
FINAL EXAMINATION
MATHEMATICS MATH 355
Analysis 4
Examiner: Professor S. W. Drury
Date: Wednesday, April 18, 2007
Associate Examiner: Professor K. N. GowriSankaran
Time: 2: 00 pm. 5: 00 pm.
INSTRUCTIONS
Attempt six questions for full cr
MATH 354
Assignment 5
Solutions
1. (i) Taking A = X, we see that B F implies that B c F. But now if A, B F then A B =
X \ (X \ A) \ B) is in F.
For (ii) we can take X = cfw_a, b, c, d and the collection of sets
F = cfw_X, cfw_a, b, cfw_b, c, cfw_c, d, cfw
MATH 354
Assignment 3
Solutions
1. (i) Clearly T f is continuous (its Lipschitz constant is 1). Also clearly T f (x) 0 for x [0, 1 ].
2
We have
1
3 1
T f (x) x
+
4 4
2
(ii) We have
x
f1 (t)2 f2 (t)2 dt
|T f1 (x) T f2 (x)| =
0
x
f1 (t)2 f2 (t)2 dt
0
1
2
|f
MATH 354
Assignment 1
Solutions
1. We have using H lders inequality with p =
o
3
2
n
and q = 3,
2
3
n
(ak bk )ck
|ak bk |
k=1
3
2
1
3
n
3
|ck |
k=1
k=1
Then use the CauchySchwarz inequality
1
2
n
n
3
2
3
2
1
2
n
3
|ak | |bk |
3
|ak |
k=1
|bk |
k=1
k=1
a
MATH 354
Assignment 2
Solutions
1. Let x U and > 0. Choose > 0 such that U (x, ) U possible since U is open. Without loss of
generality < . Since A is dense in X, there exists a A U (x, ) U U (x, ) Hence x is in the
closure of U A. This is true for every
MATH 354
Assignment 4
Solutions
1. Let fn A and suppose that fn f uniformly on [0, 1] as n . Since fn A, there exists
1
2
2
tn [0, 3 ] and sn [ 3 , 1] such that fn (tn ) = fn (sn ). Using the compactness of [0, 1 ] and [ 3 , 1], there
3
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McGill University
MATH354, Fall 2014
Midterm Test Solutions
1.
(i) Dene the term contraction mapping.
(ii) State, but do not prove the Contraction Mapping Theorem. Do not forget to include the uniqueness
statement,
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