A question on Hölder inequality

Question

Let $p, q > 1$ such that $\frac{1}{p} + \frac{1}{q} = 1$. Then $$|\sum\limits_{i = 1}^n x_i y_i| \leq ||x||_p ||x||_q, \;\; \forall x, y \in \mathbb{R}^n.$$ I have to prove it considering $$u = \frac{x}{||x||_p} \;\; \text{and} \;\; v = \frac{y}{||y||_q}$$ and using $\bf{Young's}$ inequality. Can someone, please, give me a hint?

Answer 1

Hint: Write $u=(u_1,\ldots,u_n)$ and $v=(v_1,\ldots,v_n)$.

1. By Young's inequality, $$|u_i \cdot v_i| \leq \frac{|u_i|^p}{p} + \frac{|v_i|^q}{q}. \tag{1}$$ Rewrite this inequality using the definition of $u$ and $v$.
2. Sum $(1)$ over $i=1,\ldots,n$. Deduce that $$\sum_{i=1}^n |u_i v_i| \leq 1.$$
3. Conclude.