Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 3 / 9G / b

Write $B(x,y)=⟨\alpha x,y⟩$. The hypothesis says $B(x,x)=0$. Expanding at $x+y$ giveswhereas expanding at $x+iy$ givesTherefore $B(x,y)=0$ for every $x,y$, soThis is the complex polarization argument for a vanishing quadratic form.

The conclusion is false over a real inner product space. On ${\mathbb{R}}^2$, the nonzero operatorsatisfies $⟨Jx,x⟩=0$ for every $x$.