Corollary 12.13.3.label Let $E$ be a complete nuclear space over $K \in \RC$, then $E$ is a projective limit of Hilbert spaces over $K$. For any Fréchet space $F$, $F$ is nuclear if and only if it is the projective limit of a sequence $\seq{H_n}$ of Hilbert spaces such that the mapping $H_{m} \to H_{n}$ is nuclear for all $1 \le m < n < \infty$.