Is this a valid operator norm?

Question

The "norm" (yet to be proved or disproved) defined for a matrix $A \in \mathbb{C}^{m\times n}$ by $\|A\|=\max_{i,j}|A_{i,j}|$. Is $\|\cdot\|$ a valid operator norm?

(I think it is. As it satisfies all three properties of being a norm.)

Answer 1

While $\Vert A\Vert$ as defined above does define a norm of $\mathbb{C}^{n\times m}$, to show that it is an operator norm this is not enough. You must demonstrate the following:

There exist vector spaces $V$ and $W$ equipped with norms $\Vert\cdot\Vert_V$ and $\Vert\cdot\Vert_W$, such that the space of bounded linear operators $A:V\to W$ is precisely $\mathbb{C}^{n\times m}$ euqipped with this "operator norm."

To get you started, you can use the obvious choices for $V$ and $W$.