In the context of Hilbert spaces, the following distance or metric is defined: $\delta(X,Y) \:= \sqrt{E(X-Y)^2}$, where $X$ and $Y$ are random variables such that $E(X^2)<\infty$ and $E(Y^2)<\infty$.
I want to show that $\delta(X,Y)$ is a distance function in the (traditional) sense that: $\delta(X,Y) \leq \delta(X,Z) + \delta(Z,Y)$, i.e. it satisfies the triangle inequality, where also $E(Z^2)<\infty$.
It seems like this should be a simple task, but for some reason I get stuck. I tried working with Jensen's inequality:
$\sqrt{E(X-Y)^2} = \sqrt{E(X-Z+Z-Y)^2} \geq \sqrt{[E(X-Z+Z-Y)]^2} = |E(X-Z+Z-Y)|,$
but that didn't help. Could someone please show how/if $\delta(X,Y) \leq \delta(X,Z) + \delta(Z,Y)$?