Definition 17.1.6 (Interpolation Spaces).label Let $\catc$ be a subcategory of normed spaces over $K \in \RC$, $(E_{0}, E_{1}), (F_{0}, F_{1}) \in \catc_{1}$ be a compatible couple in $\catc$, and $E, F \in \catc$, then $E$ and $F$ are interpolation spaces with respect to $(E_{0}, E_{1})$ and $(F_{0}, F_{1})$ if

1. (1)

   $E$ is an intermediate space between $E_{0}$ and $E_{1}$.

2. (2)

   $F$ is an intermediate space between $F_{0}$ and $F_{1}$.

3. (3)

   For any $T \in \text{Mor}_{\catc_1}((E_{0}, E_{1}); (F_{0}; F_{1}))$, $T|_{E}\in \text{Mor}_{\catc}(E; F)$.