## Problems

9.1 If x_{i},y_{i}, i = 1,2,…,l, are real numbers, then prove the Cauchy-Schwarz inequality:

${\left(\sum _{i=1}^{l}{x}_{i}{y}_{i}\right)}^{2}\le \left(\sum _{i=1}^{l}{x}_{i}^{2}\right)\left(\sum _{i=1}^{l}{y}_{i}^{2}\right).$

9.2 Prove that the ℓ_{2} (Euclidean) norm is a true norm, that is, it satisfies the four conditions that define a norm.

Hint

To prove the triangle inequality, use the Cauchy-Schwarz inequality.

9.3 Prove that any function that is a norm is also a convex function.

9.4 Show Young’s inequality for nonnegative real numbers a and b,

$ab\le \frac{{a}^{p}}{p}+\frac{{b}^{q}}{q},$

for $\infty >p>1$ and $\infty >q>1$ such that

$\frac{1}{p}+\frac{1}{q}=1.$

9.5 Prove Holder’s inequality for