Let $H$ be a Hilbert space and let $x,y\in H$, the Cauchy-Schwarz's inequality says that
$$|\langle x,y \rangle| \leq \|x\|\|y\|.$$
Now, in Page 15-16 of Paul R. Halmos's book, it is written that
"Cauchy-Schwarz's inequality has an interesting generalization. If $\{x_j\}$ is a non-empty, finite-family of vectors, and if $\gamma_{jk}=\langle x_j,x_k\rangle$, then the determinant of the matrix $[\gamma_{jk}]$ is non-negative; it vanishes if and only if the $x_j$'s are linearly dependent."
I do not understand this sentence. How to verify this? And how does it generalize the Cauchy-Schwarz's inequality?
Any help will be appreciated!
