cauchy-schwart inequality

definition

• for any 2 vectors $x,y\in R^n$: $|x^{T}y|\le ||x|{|}_2.||y|{|}_2$
• Generalize:
  • $|x^{T}y|\le ||x|{|}_p||y|{|}_q,\mathrm{where}\frac{1}{p}+\frac{1}{q}=1$
  • ”=” holds when $\frac{|x^{T}y|}{||x|{|}_p.||y|{|}_q}=1⟺x=\alpha y$