Question:
Let $\mathbf u, \mathbf v \in \Bbb R^n$, and let $a,b \in (1,\infty)$ such that $\frac 1a + \frac 1b = 1$. Prove that
$$||\mathbf u \mathbf v ||_1 \leq ||\mathbf u ||_a \cdot ||\mathbf v ||_b$$
where $\mathbf u \mathbf v \in \Bbb R^n$ is the component-wise product of the two vectors, and $|| \cdot ||_p$ represents the $p$-norm (although $p$ need not be an integer).
Attempt:
To be honest, I have no idea where to start. At first glance, it seems like an application of the Cauchy-Schwarz inequality, but if you put $a=b=2$, you get
$$\sum_{i=1}^n|u_iv_i| \leq \bigg(\sum_{i=1}^n|u_i|^2 \bigg)^{1/2} \bigg(\sum_{i=1}^n|v_i|^2 \bigg)^{1/2}$$
which is in fact a stronger statement than the Cauchy-Schwarz inequality with inner product being the standard dot product.
And then I'm stuck.
Any help would be much appreciated. Thanks!