A general way to get a matrix norm is inducing it from a vector norm.
The question is, can the Hilbert-Schmidt norm obtained this way?
As a norm general question, is every matrix norm obtained this way?
The crucial condition is $\|XY\| \leq \|X\|\,\|Y\|$.
The Hilbert Schmidt norm cannot be obtained as an induced norm. Of the Schatten-$p$ norms, only the $\infty$ norm (which is also the induced $2$-norm) is an induced norm.
A quick way to see that the Hilbert Schmidt norm is not induced by any matrix norm is to note that the identity matrix $I$ satisfies $\|I\| > 1$.
