I want to prove the following: If we restrict ourselves to the category $\mathcal{B}$(banach spaces), then every interpolation pair $(E,F)$ with respect to the compatible couples $(E_0,E_1),(F_0,F_1)$ is an uniform interpolation couple. Im following the proof of J. Bergh, which states that:
We consider the set of morphisms between $(E_0,E_1)$ and $(F_0,F_1)$ such that $T$ is a morphism between $E$ and $F$. We denote this space as $\mathcal{T}_1$ with the norm $$ \operatorname{max}\left(||T||_{\mathcal{L}(E,F)},||T||_{\mathcal{L}(E_0,F_0)},||T||_{\mathcal{L}(E_1,F_1)}\right),$$ and as $\mathcal{T}_2$ with the norm $$ \operatorname{max}\left(||T||_{\mathcal{L}(E_0,F_0)},||T||_{\mathcal{L}(E_1,F_1)}\right). $$ It is immediate to see that $\mathcal{T}_1$ and $\mathcal{T}_2$ are Banach spaces. Furthermore, the identity map $I:\mathcal{T}_1\to \mathcal{T}_2$ is a linear, bounded, and bijective operator. By the Banach isomorphism theorem, the inverse identity operator $I^{-1}:\mathcal{T}_2\to \mathcal{T}_1$ is bounded. It implies that $$||T||_{\mathcal{L}(E,F)}\leq \operatorname{max}\left(||T||_{\mathcal{L}(E,F)},||T||_{\mathcal{L}(E_0,F_0)},||T||_{\mathcal{L}(E_1,F_1)}\right)\leq C \operatorname{max}\left(||T||_{\mathcal{L}(E_0,F_0)},||T||_{\mathcal{L}(E_1,F_1)}\right)$$ It the last inequality where im having troubles, i don't see where it comes and how is related to the fact that $I^{-1}$ is bounded. Any help will be very appreciated
Saying that $I^{-1}$ is bounded is saying there exists some $C>0$ such that $$ \|T\|_{\mathcal{T}_1}=\|I^{-1}(T)\|_{\mathcal{T}_1} \le C\|T\|_{\mathcal{T}_2} $$ But $$ \begin{align} \|T\|_{\mathcal{T}_1}&=\operatorname{max}\left(||T||_{\mathcal{L}(E,F)},||T||_{\mathcal{L}(E_0,F_0)},||T||_{\mathcal{L}(E_1,F_1)}\right)\\ \|T\|_{\mathcal{T}_2}&=\operatorname{max}\left(||T||_{\mathcal{L}(E_0,F_0)},||T||_{\mathcal{L}(E_1,F_1)}\right) \end{align} $$