I am trying to manipulate the upper bound below using Cauchy Schwarz, so that the norm of the vector a, $||a||$, appears on the right hand side ($a_i$ itself is a scalar, $a$ is the vector). Which I think is possible with some manipulations, but I can not see it at the moment. I have the following:
$$ \sum_i a_i ^2 b^2_{ij} \leq \left(\sum_i (a_i^2)^2 \sum_i (b^2_{ij})^2 \right)^{\frac{1}{2}}$$
I'd like to have a bound of $||a||^2 $ or $||a||$ on the right hand side, where I'm less concerned about the expression/bound involving $b_{ij}$.
Any ideas?