I just wanted to clarify my understanding of the sub-multiplicative property of the matrix operator norm.
On this Wikipedia page it says for any matrix:
$ \|AB\|_{\alpha, \gamma} \leq \|A\|_{\beta, \gamma} \|B\|_{\alpha,\beta} $
But then says for square matrices:
$\|AB\|_{\alpha,\alpha} \leq \|A\|_{\alpha,\alpha}\|B\|_{\alpha,\alpha}$
I do not understand the reason for this extra section for square matrices, is this not just a special case of what was written previously?
In addition, using this can we then say that for rectangular matrices $A$ and $B$. And for the spectral norm $\|\cdot\|_2$, i.e the induced norm from using the 2-norm for both vector spaces, that:
$\|AB\|_2 \leq \|A\|_2 \|B\|_2$