Suppose that $q: V\times V \to \mathbb{C}$ is a bounded quadratic form, define
$$ \begin{equation} \tilde q(\phi,\psi) = \frac{1}{4} [q(\phi + \psi) -q(\phi - \psi) + iq(\phi + i\psi) - iq(\phi - i\psi)] \end{equation} $$
by the polarization identity. It follows that $\tilde q$ is sesquilinear. I want to show that $\tilde q$ is bounded: $|\tilde q(\phi,\psi)|\leq c||\phi||\cdot||\psi||$ for some finite constant $c$.
I was thinking about showing this using the triangle inequality:
$$ \begin{equation} |\tilde q(\phi,\psi)| \leq \frac{1}{4} [||q(\phi + \psi)|| + ||q(\phi - \psi)|| + ||q(\phi + i\psi)|| + ||iq(\phi - i\psi)||] \end{equation} $$
If we assume $\phi$ and $\psi$ are both normal vectors, then $\phi + \psi, \phi - \psi, \phi + i\psi, \phi - i\psi$ should all have the maximum norm of $\sqrt 2$. Then it follows that $|\tilde q(\phi,\psi)| \leq 2||q||$. However, there are two questions I'm not certain:
- I don't think I used the fact that $q$ is a bounded quadratic form;
- I didn't derive the form $|\tilde q(\phi,\psi)|\leq c||\phi||\cdot||\psi||$
Am I on the right track? How can I address these concerns? Thanks so much for the help:)
PS: Here's the definition I found of bounded quadratic form: a quadratic form $q$ is bounded if there exists a constant $c$ such that $|q(\phi)|\leq c||\phi||^2$ for all $\phi\in V$.