I wonder if the $2$-norm or spectral norm is also submultiplicative for non-square matrices, i.e.,
$$\| A B \|_2 \leq \| A \|_2 \cdot \| B \|_2$$
if the number of columns of $A$ coincides with the number of rows of $B$. In the literature I can only find a statement about square matrices. Thanks a lot for any remarks.
I take it that if $A$ is $m\times n$ then $||A||_2$ is the norm of $A$ as a map from $\mathbb R^m$ to $\mathbb R^n$, where both spaces have the Euclidean norm? If so then this is obvious; it's trivial that operator norms are submultiplicative: $$||STx||\le||S||\,||Tx||\le||S||\,||T||\,||x||,$$so $||ST||\le||S||\,||T||$.