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Holders Inequality

When proving Holder's inequality, why do we assume $\|x_i\|_p=1=\|y_i\|_q$, where $\frac{1}{p}+ \frac{1}{p}=1$.

$\|x_i\|_p=(\sum x_i^p)^{\frac{1}{p}}$.

Can it not be proven if the equality to unity condition is not taken?


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