Cauchy-Schwarz and Triangle Inequalities for the Mean-Square Norm

Let $(\Omega,\mathcal{F},P)$ be a probability space and let $X$ and $Y$ be square-integrable random variables on it, with the mean-square inner product $\langle X,Y\rangle_{2}=\mathbb{E}[XY]$ and norm $\lVert X\rVert_{2}$ of the same definition. Then:

1. (Cauchy-Schwarz inequality)

2. (Triangle inequality)