October 23, 2009
1
MATH 3001
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Please check that you have both pages of this test.
If you wish to use a result not proved during lectures, you must prove it here.
Please place your answers in the spaces provided. You may continue an
SUGGESTED PROBLEMS 5MATH 3001
Suppose (X, d) is a metric space.
1. Suppose cfw_fn nN and cfw_gn nN are sequences of functions in C ([0, 1])
which converge to f C ([0, 1]) and g C ([0, 1]) respectively in the
L norm (i.e converge uniformly). Prove that cfw
SUGGESTED PROBLEMS 7MATH 3001
Suppose (X, T ) is a topological space and A X . Use the denitions of
closure, interior and boundary given during lectures.
1. Prove that A C (A) Bdry(A).
2. Prove that A Bdry(A) = A. (Hint: Use the previous exercise.)
3. Sup
SUGGESTED PROBLEMS MATH 3001
Suppose (X, d) is a metric space and A X . Use the denitions of
closure, interior and boundary given during lectures.
1. Prove that if A is connected then A is connected. (Hint: If U1 is open
and a U1 A then there is some > 0
SUGGESTED PROBLEMS MATH 3001
Suppose (X, d) is a metric space. Use the denitions of closure, interior
and boundary given during lectures.
1. Suppose X is complete and A X . Prove that A is complete if and
only if A is closed.
2. Prove that (r, r) R is tot
SUGGESTED PROBLEMS MATH 3001
Suppose (X, d) is a metric space.
1. Suppose X is compact and f : X X is continuous. Prove that f
is uniformly continuous, i.e. for every > 0 there exists > 0 such
that d(x, y ) < implies that d(f (x), f (y ) < . (Hint: This i
SUGGESTED PROBLEMS 6MATH 3001
Suppose (X, T ) is a topological space and A X . Use the denitions of
closure, interior and boundary given during lectures.
1. Prove that a A if and only if every open set B containing a satises
B A = . (Hint: B A = C (B ) A.
SUGGESTED PROBLEMS 4MATH 3001
Suppose (X, d) is a metric space.
1. Suppose A X and a X . Prove that a is a limit point of A if and
only if there exists a sequence cfw_an nN A with an = a for all n N
such that lim an = a.
2. Suppose A X . Prove that A is c
SUGGESTED PROBLEMS 2MATH 3001
1. Let 1 and be the norms on C ([0, 1]) introduced in the lectures. Answer the following questions with justication. You may use
theorems from rst-year calculus without proof.
c f
(a) Does there exist c > 0 such that f
1
(b)
SUGGESTED PROBLEMS 1MATH 3001
1. Let C0 (R) be the set of continuous functions f : R R such that
limx f (x) = 0 and limx f (x) = 0. You may take for granted
that this set forms a vector space.
(a) Prove that supxR |f (x)| exists (i.e. is not innite) for e
SUGGESTED PROBLEMS 3MATH 3001
Suppose (X, d) and (Y, d ) are metric spaces
1. Suppose f : X Y and d is a metric on Y which is equivalent to d .
(a) Suppose cfw_an nN is a sequence in Y and a Y . Prove that
limn an = a with respect to d if and only if limn