Machine Learning

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 ℓ₂ (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