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
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1. (a) (8 marks) Explain the following terms:
i. Metric space
ii. Bounded set
iii. Cauchy sequence
(b) (6 marks) Prove that if A is a closed set in a metric space (X, d) then the complement X\A is an open set in (X, d).
(c) (6 marks) Which of the following sets are open and which are closed in Euclidean
space R3 ? Explain your answers.
i. {(x, y, z) ∈ R3 : z < y 2 + (x − 1)2 − 1 and xy > 0}
ii. {(x, y, z) ∈ R3 : z = 2x + 3y + 1 or x2 + y 2 = 4}
2. Let T : X → Y be a mapping between metric spaces (X, d) and (Y, d0 ).
(a) (5 marks) What does it mean to say T : X → Y is
i. a continuous mapping?
ii. an isometry?
(b) (5 marks) Prove that if T : X → Y is continuous then the preimage of every open
set in Y is an open set in X.
(c) (10 marks) State and prove Banach’s Fixed Point Theorem.
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3. (a) (3 marks) Define the operator norm k.kop for a linear operator T : X → Y between
normed vector spaces (X, k.kX ) and (Y, k.kY ).
(b) (5 marks) Prove that the operator norm is submultiplicative. (i.e. that
kST kop ≤ kSkop kT kop
for operators S : Y → Z and T : X → Y ).
(c) (6 marks) What is the spectral norm for an n × n matrix?
Compute the spectral norm of the following matrix.


1 2


1 1
(d) (6 marks) Let T : X → X be a continuous linear operator on a Banach space
(X, k.k). Prove that if kT kop < 1 then I − T is invertible. (Here I denotes the
identity operator on X).
4. (a) (6 marks) What is a topological space? Give an example of a topological space
which is not metrizable.
(b) (6 marks) Prove that if (X, τ ) is a connected topological space then the empty
set ∅ and X are the only subsets of X which are both open and closed.
(c) (8 marks) Prove that if (X, τ ) is a compact metric space then every sequence in
X has a convergent subsequence.
c UNIVERSITY OF DUBLIN 2012