If $E$ is a Fréchet space and $(p_j)$ an increasing sequence of semi-norms on $E$ defining the topology of $E$ we have $$ E=\underset{\longleftarrow}{\lim}\,\widehat E_{p_j}, $$ where $\widehat E_{p_j}$ is the Hausdorff completion of $(E,p_j)$ and $\widehat E_{p_{j+1}}\to\widehat E_{p_j}$ the canonical morphism. Here $\widehat E_{p_j}$ is a Banach space for the induced norm $\widehat p_j$

A Fréchet space $E$ is said to be nuclear if the topology of $E$ can be defined by an increasing sequence of semi-norms $p_j$ such that each canonical morphism $\widehat E_{p_{j+1}}\to\widehat E_{p_j}$ of Banach spaces is nuclear.

Recall that a morphism $f\colon E\to F$ of complete locally convex spaces is said to be nuclear if $f$ can be written as $$ f(x)=\sum\lambda_j\xi_j(x)y_j $$ where $(\lambda_j)$ is a sequence of scalars with $\sum|\lambda_j|<+\infty$ $\xi_j\in E'$ an equicontinuous sequence of linear forms and $y_j\in F$ a bounded sequence.