How would I prove that for standard inner product on $\mathbb{R^2}$:
where $\langle v,w\rangle := \sum_{i=1}^2 v_i \cdot w_i$,
$$(v_1w_1 + v_2w_2)^2 \le (v_1^2 +v_2^2)(w_1^2+w_2^2)$$
Hint we were given: $$f(v_1, v_2, w_1, w_2) = (v_1^2 +v_2^2)(w_1^2+w_2^2) - (v_1w_1 + v_2w_2)^2$$
HINT
We need to find te global minimum of
$$f(v_1, v_2, w_1, w_2) = (v_1^2 +v_2^2)(w_1^2+w_2^2) - (v_1w_1 + v_2w_2)^2$$