Definition 14.7.15 (Quadratic Function).label Let $H$ be a complex Hilbert space and $q: H \to \complex$, then $q$ is quadratic if:

1. (1)

   For each $\lambda \in \complex$ and $\xi \in H$, $q(\lambda \xi) = |\lambda|^{2} q(\xi)$.

2. (2)

   (Parallelogram Law) For each $\xi, \eta \in H$,

   \[q(\xi + \eta) + q(\xi - \eta) = 2q(\xi) + 2q(\eta)\]

   
   

and bounded if

1. (3)

   There exists $M \ge 0$ such that $|q(\xi)| \le M\norm{\xi}_{H}^{2}$ for all $\xi \in H$.