Is the Frobenius matrix norm induced?

In this paper we answer the question of whether (or not) there exist normed input-output vector spaces that induce the Frobenius matrix norm. Specifically, using the notion of dual norms we show that, up to a scalar multiple, the maximum singular value is the only unitarily invariant induced norm. As a special case of this result, it follows that the Frobenius matrix norm is not induced.