EDB — 10M

[10M] Let be given $p,q\in [1,\infty ]$ such that $1/p+1/q=1$  ¹ and $x,y\in {ℝ}^n$; show the Hölder inequality in this form

In what cases is there equality?

Tips: Fix $x_i,y_i\ge 0$. For the cases with $p,q<\infty$ you can:

• use Young inequality ([194]↺↻ or [1V7]↺↻);

• use Lagrange multipliers;

• start from the case $n=2$ and set $x_2=t{x}_1$ and $y_2=a{y}_1$; then, for cases $n\ge 3$ use induction.