I'm reading about functional analysis and I found the definition of the operator norm, if you have $(X,\|\|_1)$ and $(Y,\|\|_2)$ normed spaces then the set $\mathcal{L}_{\|\|_1,\|\|_2}(X,Y) := \{T:X \to Y \text{ linear }: \sup\{ \|T(v)\|_2: \|v\|_1 = 1 \} < \infty \}$ has a norm defined by $\|\|_1$ and $\|\|_2$ and son on. My questions are:
If $\mathcal{L}_{\|\|_1,\|\|_2}(X,Y) = \mathcal{L}_{\|\|'_1,\|\|_2}(X,Y)$ then, can I ensure that $\|\|_1$ and $\|\|'_1$ are equivalents? Note that this generalizes the fact that all the norms in $\mathbb{R}^n$ are equivalents, because any linear operator is continuos with any norm in $\mathbb{R}^n$. Similarly with the other side,
If $\mathcal{L}_{\|\|_1,\|\|_2}(X,Y) \subseteq \mathcal{L}_{\|\|'_1,\|\|_2}(X,Y)$, can I say something? And like before with the other side,
If I have a subspace $Z$ of $\mathbb{L}(X,Y)$ then there exist norms such that $Z = \mathcal{L}_{\|\|_1,\|\|_2}(X,Y)$.
I apologize if my questions are not interesting, thank for your help.