# Frobenius norm and submultiplicativity

I read (page 8 here) that if $A$ and $B$ are rectangular matrices so that the product $AB$ is defined, then $$(1)\quad||AB||_F^2\leq ||A||_F^2||B||_F^2$$

Does that mean that the inequality above also holds when the number of rows of $A$ is larger than the number of columns of $B$? The justification (Cauchy Swartz):

$$||AB||_F^2=\sum_{i=1}^n\sum_{j=1}^k(a_i^\top b_j)^2\leq \sum_{i=1}^n\sum_{j=1}^k||a_i||_2^2||b_j||^2_2=||A||_F^2||B||_F^2$$

does not require $k$ (the number of columns of $B$) to equal $n$ (the number of rows of $A$). Intuitively, you could also add imaginary columns of 0's to $B$, so I can believe the claim. On the other hand, in other places I only see $(1)$ claimed for matrix of the same size and have had a hard time finding it claimed for the more general case (where $A$ $B$ are merely multiplicative) online.

Indeed, the only requirement to have the inequality that you wrote for the Frobenius norm and for arbitrary matrices is that the product $AB$ is defined.