# Inequality in infinite Hilbert space

Let H an infinite-dimensional Hilbert space in $\mathbb{R}$

If $x_1, x_2, \ldots x_n \in H$, how to prove:

$\sum_{1\leq i,j\leq n} {\lvert\lvert x_i - x_j \rvert\rvert}^2 \leq \sum_{1\leq i,j\leq n} ({\lvert\lvert x_i \rvert\rvert}^2 + {\lvert\lvert x_j \rvert\rvert}^2)$

• I'm completely blanking on how this isn't just the triangle inequality. – Titus Jan 14 '16 at 1:59

The triangle inequality yields only $\|x_i - x_j\|^2 \le (\|x_i\| + \|x_j\|)^2 \le \|x_i\|^2 + 2 \|x_i\| \|x_j\| + \|x_j\|^2 \le 2 (\|x_i\|^2 + \|x_j\|^2)$...
• $(|x|+|y|)^2$ is not equal to $|x|^2+2\langle x,y \rangle+|y|^2$. – wj32 Jan 17 '16 at 11:30