Page 368 - Advanced Linear Algebra
P. 368
352 Advanced Linear Algebra
Exercises
1. Prove that the sup metric on the metric space *´ Á µ of continuous
functions on ´ Á µ does not come from an inner product. Hint: let ²!³ ~
and ²!³ ~ ²! c a³°² c a and consider the parallelogram law.
³
2. Prove that any Cauchy sequence that has a convergent subsequence must
itself converge.
3. Let be an inner product space and let and be subsets of . Show
=
(
)
=
that
a ) ( ) ¬ ) (
b ( ) is a closed subspace of =
c ) ´ cspan ( ² ³ µ ~ (
4. Let = be an inner product space and : = . Under what conditions is
: ~ : ?
5. Prove that a subspace of a Hilbert space / is closed if and only if
:
:~ : .
6. Let be the subspace of consisting of all sequences of real numbers
M
=
with the property that each sequence has only a finite number of nonzero
terms. Thus, = is an inner product space. Let 2 be the subspace of =
consisting of all sequences %~²% ³ in = with the property that
'%° ~ . Show that 2 is closed, but that 2 £ 2. Hint: For the latter,
show that 2 ~ ¸ ¹ by considering the sequences " ~ ² ÁÃÁc Á ó ,
where the term c is in the th coordinate position.
7. Let E be an orthonormal set in . If % ~ " converges,
' ~ ¸"Á "Á Ã ¹
/
show that
B
%
)) ~ ( (
~
8. Prove that if an infinite series
B
%
~
converges absolutely in a Hilbert space / , then it also converges in the
sense of the “net” definition given in this section.
9. Let ¸ 2¹ be a collection of nonnegative real numbers. If the sum
on the left below converges, show that
~ sup
1 finite
2 12 1
10. Find a countably infinite sum of real numbers that converges in the sense of
partial sums, but not in the sense of nets.
11. Prove that if a Hilbert space / has infinite Hilbert dimension, then no
Hilbert basis for is a Hamel basis.
/
12. Prove that M²2³ is a Hilbert space for any nonempty set .
2
