Im prooving the inequality: $\|AB\|_F \leq \|A\|_2 \|B\|_F$. To prove this I need to know, if the following is true:
Lets $B_{n \times r}~=~(\mathbf{b_1}, \ldots, \mathbf{b_r})$ is a matrix, $\mathbf{b_1}$, $\ldots$, $\mathbf{b_r}$ are vectors $n \times 1$, then \begin{equation*} \|B\|_2^2~=~\|\mathbf{b_1}\|_2^2~+\ldots~+~\|\mathbf{b_r}\|_2^2. \end{equation*}
Lets $A_{m \times n}, B_{n \times r}~=~(\mathbf{b_1}, \ldots, \mathbf{b_r})$ are matrices, $\mathbf{b_1}$, $\ldots$, $\mathbf{b_r}$ are vectors $n \times 1$, then \begin{equation*} \|AB\|_F^2~=~\|A\mathbf{b_1}\|_F^2~+\ldots~+~\|A\mathbf{b_r}\|_F^2. \end{equation*}
If these two equations are true, then I can finish the proof. In other case, its bad. Can anybody say me, whether they are true or not and in the case they are true, why?
Thank you very much. Eva