[Math] About Cauchy–Schwarz inequality

Question

For the vectors $x$ and $y$, the Cauchy–Schwarz inequality reads
$$
|x\cdot y|\leq||x||\cdot||y||
$$
Does this inequality only hold for 2-norm? Or for any norms?

Thanks in advance.

Answer 1

With finite sequences $x_i$ and $y_i$, $1\le i\le N$ (assumed positive, or add absolute values to the $x_i$s and $y_i$s) you have a generalized Rogers-Hölder's inequality: for $u,v, w>0$ and $1/u+1/v\le 1/w$:

$$\Big(\sum_1^N (x_i y_i)^w\Big)^\frac{1}{w} \le \Big(\sum_1^N (x_i )^u\Big)^\frac{1}{u} \Big(\sum_1^N (y_i )^v\Big)^\frac{1}{v} $$

See for instance P. S. Bullen, Handbook of Means and Their Inequalities, 2003, p. 188. I called them Rogers-Hölder from L. Maligranda, “Why Hölder’s inequality should be called Rogers' inequality? ” Math. Inequal. Appl., vol. 1, no. 1, pp. 69–83, 1998.

You recover your case with $w=1$, and the inequality on $u,v,w$ offers you different options.