Definition 17.2.1 (Calderón Space).label Let $S = \bracs{z \in \complex| \text{Re}(z) \in (0, 1)}$ and $(E_{0}, E_{1})$ be a compatible couple of Banach spaces over $\complex$, then the Calderón space $\cf(E_{0}, E_{1})$ is the Banach space of functions $f: \ol S \to E_{0} + E_{1}$ such that:

1. (1)

   $f$ is holomorphic on $S$.

2. (2)

   $f$ is continuous on $\ol S$.

3. (3)

   For each $t \in \real$, $f(it) \in E_{0}$, and $\lim_{|t| \to \infty}\norm{f(it)}_{E_0}= 0$.

4. (4)

   For each $t \in \real$, $f(1 + it) \in E_{1}$, and $\lim_{|t| \to \infty}\norm{f(1 + it)}_{E_1}= 0$.

equipped with the norm