Math 442
Exercise set 1
1. If H" ." H# .# are metric spaces, B H" , and 0 H" H# a map, prove
that 0 is continuous at B H" in the sense of 1.5 if and only if, for any metrically open set
Y containing 0 B , there is a metrically open set Z containing B and
Math 442
Exercise set 2, 2008 sketch solutions
1.
(a)
cl0 E is closed in H# . Thus, by 2.5, 0
0
and so, by the definition 2.3, 0
"
cl0 E 0
"
"
"
cl0 E is closed in H" . But
0 E E ,
cl0 E cl E , and so 0 clE cl0 E .
[2]
(b) Let Y be open in H$ . Then 1 " Y
Math 442
Exercise set 5
1.
Let I be a complex inner product space. Prove the so-called polarization identity:
aB C I
%B C lB
C l#
lB
C l#
3lB
3C l#
lB
3C l# .
2. Deduce from the previous question that, if I is a real normed space for which the
Apollonian
Math 442
Exercise set 1, 2008 sketch solutions
cfw_Well, they are actually far fuller solutions than Id ever expect anyone to present.
1. Suppose, firstly, that 0 is metrically continuous at B H" . The definition (1) of 1.5
may be rephrased as
a%
!b$
!
C
Math 442
Exercise set 4 sketch solutions
8
1. Take a sequence B8 8 03 3 8 , where each B8 -! and the sequence is
Cauchy in -! . This means that, for given % ! , there is some R % such that
7 8 R % sup03
7
03 3
8
% . For each specific index 3,
7 8 R % 03
Math 442
Exercise set 3, 2008 sketch solutions
1. Certainly the formula makes sense (since 0 and 1 are continuous on ! " , the
integral of 0 1 is defined) and it is obviously linear in 0 . So it is a linear functional on
G G! " . It only remains to show t
MATH 442
Analysis II: Topics in Analysis
2008
0. Introduction.
As it says in the Prospectus, there are several possible topics that might be treated in this
course, and I cannot pretend to have made an authoritative selection. But I hope that all the
matt
MATH 442
Test 2
2008
1. Let be a Banach space, and suppose that is an inner product on which
is bounded in the sense that, for some positive constant ,
.
(1)
(The norm induced by is not supposed to be the same as ). Prove that, if is a
Hilbert space with
MATH 442
Test 1
2008
1. Suppose that is a complete uncountable metric space, and a
continuous map. For each , define , and set inductively
for each . Suppose that is convergent in for
every .
(a) Show that must have at least one fixed point.
(b) Give an
Math 442
Exercise set 6, 2008
In various Hilbert spaces, there are standard orthonormal bases, which arose in many
cases from important problems.
One of the most famous is the basis in P# " " formed by the Legendre polynomials,
also called Legendre coeffi