If $V$ is a finite dimensional Hilbert space for any vector $x \in V$ and endomorphism $A$, the function $ Q(x) = \langle Ax, x \rangle $ defines a quadratic form on $V$. Now, I would like to show that $Q$ is continuous. The main thought I have about this is that it is very similar to the (analytic) Riesz representation theorem. If $Q$ were linear, we would be done for Riesz guarantees that all linear forms are continuous. However, $Q$ is actually bilinear so Riesz doesn't apply. Also, I'm aware of the machinery that one can use to solve $n$-linear continuity problems such as this one, i.e., a multilinear map is continuous iff it is bounded iff int is continuous at $0$, etc., and these are the same techniques that one uses with linear maps. This brings me to my question:
Is there a simple way to demonstrate that the function $Q$ as defined above is continuous? Is there perhaps an elementary way to extend Riesz in order to apply it to bilinear functions?