Proof, [Theorem III.7.4, SW99]. Firstly, a setup about auxiliary spaces and subspaces is required. Let $U \in \cn_{E}(0)$ be convex and circled, then the composition of the inclusion map $\iota: F \to E$ and the canonical projection $\pi_{U}: E \to E_{U}$ factors through $F_{U \cap F}$ as follows:

where $\widehat \pi_{U}$ is an isometric embedding. As a result, the factored map $\widehat \pi_{U}: F_{U \cap F}\to E_{U}$ extends to an isometric embedding on the completions:

which enables identifying $\widehat F_{U \cap F}$ as a closed subspace of $\widehat E_{U}$.

To start the proof, let $U \in \cn_{E}(0)$ be a given 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. By prior discussion, the following diagram commutes:

Thus the induced map $\widehat \pi_{U \cap F}: \widehat F_{V \cap F}\to \widehat F_{U \cap F}$ corresponds to the restriction of $\widehat \pi_{U}$ to $\widehat F_{V \cap F}$. Since $\widehat \pi_{U}$ is nuclear, there exists $\seq{\phi_n}\subset E_{V}^{*}$ and $\seq{y_n}\subset \widehat E_{U}$ such that

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

Now, using Theorem 12.13.2, further assume without loss of generality that $\widehat E_{U}$ is a Hilbert space. Let $P: \widehat E_{U}\to \widehat F_{U \cap F}$ be the orthogonal projection of $\widehat E_{U}$ onto $\widehat F_{U \cap F}$, then

with

Therefore the induced map $\widehat \pi_{U \cap F}$ is nuclear, and $F$ is a nuclear space.$\square$