## Problems

9.1 If xi,yi, 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 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 $∞>p>1$ and $∞>q>1$ such that

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

9.5 Prove Holder’s inequality for

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