Definition 26.4.1 (Codomain of Derivatives).label Let $E, F$ be TVSs over $K \in \RC$, $\sigma \subset \mathfrak{B}(E)$ be a covering ideal, and $L^{(0)}_{\sigma}(E; F) = F$. For each $n \in \natp$, inductively define
\[L^{(n)}_{\sigma}(E; F) = L(E; L^{(n-1)}_{\sigma}(E; F)) \subset B_{\sigma}^{n}(E; F)\]
and equip it with the $\sigma$-uniform topology, then under the identification
\[I: L^{(n)}_{\sigma}(E; F) \to B_{\sigma}^{n}(E; F) \quad I\lambda(x_{1}, \cdots, x_{n}) = \lambda(x_{1})\cdots(x_{n})\]
the space $L^{(n)}_{\sigma}(E; F)$ is a subspace of $B_{\sigma}^{n}(E; F)$.
Proof. By Proposition 10.13.3.$\square$
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