Math 318 HW #10
Due 3:00 PM Friday, April 29
Reading:
Wilcox & Myers §32–37.
Problems:
1. Exercise 34.36.
Comment: The unit ball not being compact is a common feature of infinite-dimensional
normed linear spaces which can be a serious nuisance. In fact, one can prove that the unit
ball in a normed linear space is compact if and only if the space is finite-dimensional.
2. Exercise 38.20. Note that Parseval’s Theorem is correctly stated in this exercise, but the
statement is incorrect in Corollary 37.10.
Comment: If you’ve ever encountered `2 , Parseval’s Theorem tells us that the L 2 norm of a
function is the same as the `2 norm of the sequence of terms of its Fourier series.
3. Exercise 38.22. Note that I’ve been using hf, gi2 in class to indicate the L 2 inner product of
f and g, whereas Wilcox & Myers use f · g to mean the same thing. Also, there’s a typo in
the statement of the problem: the first terms in the Fourier series for f and g should be a20
and
α0
2 ,
and not
a20
2
and
α20
2 .
Comment: This is just the L 2 analogue of the usual way we compute inner products (dot
products) in Rn by summing the products of the components.
4. Exercise 38.27.
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