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Machine Learning by Sergios Theodoridis

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Problems

9.1 If xi,yi, i = 1,2,…,l, are real numbers, then prove the Cauchy-Schwarz inequality:

i=1lxiyi2i=1lxi2i=1lyi2.

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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,

abapp+bqq,

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for >p>1 and >q>1 such that

1p+1q=1.

9.5 Prove Holder’s inequality for

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