Let $V,W$ be normable topological vector spaces over $\mathbb{F}$.
Let $C(V,W)$ be the set of continuous linear transformations $T:V\rightarrow W$.
Let $||\cdot||_V, ||\cdot||_W$ be norms on $V,W$ inducing the given topologies on $V,W$ respectively, and $||\cdot||_{op}$ be the operator norm on $C(V,W)$ induced by $||\cdot||_V$ and $||\cdot||_W$.
Then, the topology on $C(V,W)$ induced by the operator norm does not depend on choices of $||\cdot||_V,||\cdot||_W$. (To be precise, the operator norm of another choice of norms $||\cdot||_1,||\cdot||_2$ induce the same topology as the original one if $||\cdot||_1$, $||\cdot||_V$ are equivalent and $||\cdot||_2$, $||\cdot||_W$ are equivalent) Hence, it deserves to be called "the standard topology on $C(V,W)$".
This makes me curious to think of two questions.
Let $||\cdot||_c$ be a norm on $C(V,W)$ inducing the standard topology on $C(V,W)$. Then, does there exist two norms $||\cdot||_V,||\cdot||_W$ on $V,W$ respectively of which the operator norm is $||\cdot||_c$?
Let $||\cdot||_V, ||\cdot||_W$ be norms on $V,W$ respectively. Then, does there exist a norm which can be manipulated from $||\cdot||_V,||\cdot||_W$, not inducing the standard topology on $C(V,W)$? That is, is the standard topology the unique topology on $C(V,W)$ in some sense?
Thank you in advance.