I need to show the following two properties of the Frobenius Norm.
(i) $\|A^{T}A\|_{F} \leq \|A\|_{F}^{2}$.
(ii) $\|AB\|_{F} \leq \|A\|_{F}\|B\|_{F}$, where $A$ is m-by-k and $B$ is k-by-n.
Now, the second proof must not use indices. Therefore, while I have found the second answer to Frobenius Norm Inequality; Spectral Radius is smaller than Frobenius Norm to be useful, I am not sure how to prove the second inequality without reference to indices. I assume the trace is important, but I'm not sure how to use it.