the Cauchy Schwarz inequality says for example $$ (a_1^2+a_2^2+a_3^2)(b_1^2+b_2^2+b_3^2) \geq (a_1 b_1+ a_2 b_2+a_3 b_3)^2$$ and with clever choices of $a$ and $b$ we can solve many problems. It can be read as saying $\cos \theta \leq 1$. It can be interpreted as quantifying the pigeonhole principle. Also it's related to the distance from a point to a line (leading to a proof). Half of real analysis is clever usages of this inequality.
Cauchy Schwartz works because it's related to Pythagoras theorem, and it extends to $n$ variables and to metric spaces. So here is my question. Let $V=\mathbb{R}^n$ with the dot product. Then $V \wedge V$ is also an inner product space with an induced inner product. How do I write the Cauchy Schwartz inequality there.
for Pythagoras theorem I have $$ |v \wedge w |^2= \sum (v_i w_j - w_i v_j)^2$$ which says the area squares of a parallelogram in $n$ space is the sum of the area squares of each projection into pairs of coordinate axes.
what is the analogue of Cauchy Schwartz?