Definition 18.1.1 (Derivatives and Remainders). Let $E, F$ be TVSs over $K \in \RC$ and $\ch(E; F), \calr(E; F) \subset F^{E}$ be vector subspaces, then $\mathcal{HR}= (\ch(E; F), \calr(E; F))$ is a pair of derivatives and remainders if

1. For any $T \in \ch$, if there exists $V \in \cn_{E}(0)$ and $r \in \calr$ such that $T|_{V} = r|_{V}$, then $T = 0$.