Definition 18.2.1 (Small). Let $E, F$ be TVSs over $K \in \RC$, $\sigma \subset B(E)$ be an upward-directed family that contains all finite sets, $r: E \to F$, and $n \in \natz$, then the following are equivalent:

1. For each $A \in \sigma$, $r(th)/t^{n} \to 0$ uniformly on $A$.

2. If $r_{t}(x) = r(tx)/t^{n}$, then $r_{t} \to 0$ as $t \to 0$ with respect to the $\sigma$-uniform topology on $F^{E}$.

3. For each $A \in \sigma$, $\seq{a_k}\subset A$, and $\seq{t_k}\subset K \setminus \bracs{0}$ with $t_{k} \to 0$ as $n \to \infty$, $r(t_{k}a_{k})/t_{k}^{n} \to 0$ as $n \to \infty$.

If the above holds, then $r$ is $\sigma$-small of order $n$.

The set $\mathcal{R}_{\sigma}^{n}(E; F)$ is the $K$-vector space of all $\sigma$-small functions of order $n$ from $E$ to $F$. For simplicity, $\mathcal{R}_{\sigma}(E; F)$ denotes $\mathcal{R}_{\sigma}^{1}(E; F)$.