I have come across a question:
Let $f$ be a complex function on a finite set $X$, i.e. $f: X \rightarrow \mathbb{C}$. Let $1\leq p \leq q \leq \infty$. Show that $||f||_{L_p}\leq ||f||_{L_q}$ and $||f||_{l_p}\geq ||f||_{l_q}$.
I am really confused by the meaning of the $L_p$ norm on a function defined on a finite set. I think the $l_p$ norm is simply:
$||f||_{l_p} = (\Sigma_{x\in X} f(x)^p )^{1/p}$
but cannot think of what the $L_p$ norm could be. Wikipedia uses an integral, but since $X$ is finite would this not just reduce to the sum above?
Background: in case this is relevant, this question comes from a course on fourier methods in combinatorics.