Definition 12.12.1 (Nuclear Operator Between Banach Spaces).label Let $E, F$ be Banach spaces, $E^{*}$ be the dual of $E$, equipped with the uniform topology, and $T \in L(E; F)$, then $T$ is nuclear if there exists $\seq{\phi_n}\subset E^{*}$ and $\seq{y_n}\subset F$ such that:

1. (1)

   For each $x \in E$, $Tx = \sum_{n = 1}^{\infty} y_{n} \dpn{x, \phi_n}{E}$.

2. (2)

   $\sum_{n \in \natp}\norm{y_n}_{F}\norm{\phi_n}_{E^*}< \infty$.

The set $N(E; F)$ is the space of nuclear operators from $E$ to $F$. For each $T \in N(E; F)$, let

then $\norm{\cdot}_{N(E; F)}$ is a norm on $N(E; F)$, and $N(E; F)$ is a Banach space.