Proof, [Theorem III.7.4, SW99]. Firstly, a setup about auxiliary spaces and quotients is required. Let $p: E \to E/F$ be the canonical projection and $U \in \cn_{E}(0)$ be a convex and circled neighbourhood, then the composition of maps $E \to E/F \to (E/F)_{p(U)}$ factors through $E_{U}$ as follows:

This extends through the completion

and yields that $\widehat{(E/F)}_{p(U)}$ is a quotient space of $\widehat E_{U}$.

To begin the proof, let $U \in \cn_{E}(0)$ be a convex and circled neighbourhood. Since $E$ is nuclear, there exists a convex and circled neighbourhood $V \in \cn_{E}(0)$ with $V \subset U$ such that the induced map $\widehat \pi_{U}: \widehat E_{V} \to \widehat E_{U}$ is nuclear. The composition of maps $\wh E_{V} \to \wh E_{U} \to \wh{(E/F)}_{p(U)}$ then factors through $\widehat{(E/F)}_{p(V)}$ as $\widehat \pi_{p(U)}$:

Since $\wh \pi_{U}: \wh E_{V} \to \wh E_{U}$ is nuclear, there exists $\seq{\phi_n}\subset E_{V}^{*}$ and $\seq{y_n}\subset \wh E_{U}$ such that

and $\sum_{n \in \natp}\norm{y_n}_{\wh E_U}\norm{\phi_n}_{E_V^*}< \infty$.

Now, using Theorem 12.13.2, further assume without loss of generality that $\wh E_{V}$ is a Hilbert space. Identify $(\widehat{E/F})_{p(V)}$ as a closed subspace of $\widehat E_{V}$, and let $P: \widehat E_{V} \to (\widehat{E/F})_{p(V)}$ be the orthogonal projection of $\widehat E_{V}$ onto $(\widehat{E/F})_{p(V)}$. This allows rewriting

where

Therefore $\widehat \pi_{p(U)}$ is nuclear, and $E/F$ is a nuclear space.$\square$