MATH 731, FALL 2008
HOMEWORK SET 3
Due Friday, September 26 at noon
In these problems, you may assume we are working over F=C.
A. (i) Let Vbe the vector space of continuous complex-valued functions with domain [0,1].
Show Vdoes not become an inner product space when we define hf , gi=R1
0f(x)g(x)dx
for f, g ∈V.
(ii) Explain how to alter (in a simple way) the definition of hf, g iin (i) to obtain an inner
product.
(iii) Verify that your altered definition does in fact yield an inner product.
B. Let h,ibe an inner product on the vector space Vand define k·k :V→Rby kvk=
phv, vi. Prove that k·k :V→Ris a norm, that is, prove the following three properties
hold.
(1) kvk>0 for v∈V, v 6= 0;
(2) kcvk=|c|kvkfor c∈F, v ∈V;
(3) kv+wk ≤ kvk+kwkfor all v, w ∈V.
Hint: You may need to use the Cauchy-Schwartz Inequality |hv , wi| ≤ kvkkwkfor all
v, w ∈V. You can assume this fact.
(Four bonus points for proving the Cauchy-Schwartz Inequality.)