I noticed recently that the Cauchy-Schwarz inequality can be derived from the positive defineteness of a quadratic form $f: \mathbf{R}^2 \times \mathbf{R}^2 \rightarrow \mathbf{R}$.
Given $\mathbf v_1$ and $\mathbf v_2$ in a generic Hilbert space $H$:
$$f(\mathbf {\lambda} ,\mathbf {\mu})=(\sum_{i=1,2} \lambda_i \mathbf v_i, \sum_{j=1,2} \mu_j \mathbf v_j)=\sum_{i=1,2,j=1,2} \lambda_i ( \mathbf v_i, \mathbf v_j) \mu_j$$
, which is just a different way to see the usual elementary proof with the discrimant.
The previous argument nevertheless produces in an easy way different inequalities for any subset of vectors $\mathbf v_1,..,\mathbf v_k$ even for $k > 2$ by considering quadratic forms in higher dimensional spaces.
I explain a bit more the strategy, as requested. For example one can write the symmetric matrix $M_{i,j}=( \mathbf v_i, \mathbf v_j), i=1...k,j=1...k$ and deduce from the positivity of the quadratic form:
$$ det M \ge 0$$
For $k=2$ this expands to CS. For $k>2$ bunch of inequalities are produced.
So my questions:
Why are these inequalities often not presented ? I never saw such an argument but I guess it should be well know ? Is there any reference ?
Is there a reason for which the inequalities with $k>2$ are less useful than CS and therefore deserve less popularity ? Maybe these inequalities are implied by CS ?
EDIT: following Angina Seng comment I found this link which follows this strategy :