UNIVERSITY OF DUBLIN
XMA
TRINITY COLLEGE
Faculty of Engineering, Mathematics
and Science
school of mathematics
SF Mathematics
SF TSM Mathematics
Trinity Term 2012
Module MA2223
Dr. D. Kitson
Page 2 of 3
XMA
1. (a) (8 marks) Explain the following terms:
i.
MA2223: SOLUTIONS TO PROBLEM SHEET 2
1. Let (X, d) be a metric space and let A be a subset of X.
(a) Prove that the closure A is the intersection of all closed sets
which contain A.
(b) Using (a) show that the closure of A is closed.
Solution: (a) Suppose
MA2223: SOLUTIONS TO ASSIGNMENT 5
1. State which of the following collections is a topology on the set X =
cfw_a, b, c.
(a) 1 = cfw_, cfw_a, cfw_a, b, c
Answer: Topology.
(b) 2 = cfw_, cfw_a, b, cfw_c, cfw_a, b, c
Answer: Topology.
(c) 3 = cfw_, cfw_a, b,
MA2223: SOLUTIONS TO PROBLEM SHEET 3
1. Let
2
be the real vector space of all square summable sequences
x = (xn ) of real numbers with norm
1
2
x
2
x2
i
=
i=1
Show that the unilateral shift operator
T :
2
2
,
(x1 , x2 , x3 , . . .) (0, x1 , x2 , x3 , . .
MA2223: SOLUTIONS TO PROBLEM SHEET 4
1. Let A and B be open sets in a normed vector space (X, . ) and let
r be a positive real number. Prove that the following sets are open
(a)
A + B = cfw_a + b : a A, b B
(b)
rA = cfw_ra : a A
Solution:
(a) We can expre
MA2223: SOLUTIONS TO ASSIGNMENT 4
1. Prove directly that the following three norms on R2 are equivalent.
(a) x
1
= |x1 | + |x2 |
(b) x
2
=
(c) x
(x1 )2 + (x2 )2
= maxi=1,2 |xi |
where x = (x1 , x2 ).
Solution: First we will show that the 1-norm and the ma
MA2223: SOLUTIONS TO ASSIGNMENT 2
1. Determine which of the following subsets of R are open.
(a) cfw_x R : 1 < x < 3
Answer: Open.
Reason: We proved in class that in a metric space every open ball
is an open set. Here we have an open ball in R with centre
MA2223: SOLUTIONS TO ASSIGNMENT 3
1. Verify that each of the following mappings are isometries on R2 .
(a) Reection.
T : R2 R2 , (x1 , x2 ) (x2 , x1 )
Let x = (x1 , x2 ) and y = (y1 , y2 ) be points in R2 . Recall that the
Euclidean metric on R2 is given
MA2223: SOLUTIONS TO PROBLEM SHEET 1
1. Show that the diameter of an open ball B(a, r) in Euclidean space
Rn is exactly 2r.
Solution: We proved in class that in any metric space the diameter of an open ball is at most twice the radius. So we know
diam(B(a