I am considering the series case. In the Holder inequality, we have $$\sum|x_iy_i|\leq\left(\sum|x_i|^p\right)^{\frac1p} \left(\sum|y_i|^q\right)^{\frac1q},$$ where $\frac1p+\frac1q=1,~p, q>1$.

In Cauchy inequality (i.e., $p=q=2$), I know that the equality holds if and only if $x$ and $y$ are linearly dependent. I am wondering when the equality holds in the Holder inequality.