Let $A$, $B$ be real $m \times n$ matrices. I want to show
$$\left\lVert A'B \right\rVert^2 \leq \left\lVert A'A \right\rVert \left\lVert B'B \right\rVert$$
where the norm is the Frobenius norm. Since $\langle A,B \rangle=Tr(AB')$ is an inner product on the space of $m \times n$ matrices Cauchy-Schwarz's inequality gives me
$$ Tr(A'A B'B) \leq \left\lVert A'A \right\rVert \left\lVert B'B \right\rVert$$
But $Tr(A'A B'B)=Tr(A B'BA')=\left\lVert A B' \right\rVert^2$ and not $\left\lVert A'B \right\rVert^2$.
It seems like only a small adjustment is needed but I can't find it!
Any help is greatly appreciated.