Fréchet spaces used in analytic coherent cohomology

A Fréchet space over $\mathbb{C}$ is a complex vector space $E$ with a Hausdorff locally convex vector-space topology generated by a separating countable family of seminorms $(p_n{)}_{n\ge 1}$ and complete for its seminorm uniformity. Explicitly, use the translation-invariant compatible metric $d(x,y)={\sum }_{n\ge 1}{2}^{-n}\min \{1,p_n(x-y)\},$ and require that this metric is complete. A seminorm is a function $p:E\to [0,\mathrm{\infty })$ satisfying $p(x+y)\le p(x)+p(y)$ and $p(cx)=\mathrm{\mid }c\mathrm{\mid }p(x)$ for $x,y\in E$ and $c\in \mathbb{C}$; separating means that for $x\mathrm{\ne }0$ some $p_n(x)>0$. Locally convex means that every point has a neighborhood basis of convex sets. One may replace the defining family by the increasing family of its finite maxima.

Completeness here always refers to this vector-space uniformity. A sequence is Cauchy for the displayed metric exactly when it is Cauchy for every defining seminorm: each individual metric summand controls that seminorm at sufficiently small distances, and conversely the geometric tail of the metric series plus finitely many seminorm estimates controls the metric. Thus a seminorm-Cauchy series used in the accompanying proofs converges in $E$. The zero vector space is included, with its unique metric. This is the usual complete metrizable locally convex Fréchet convention.

Jean-Pierre Demailly, Complex Analytic and Differential Geometry, Chapter IX §1.A–1.B

Henri Cartan, Un théorème de finitude, Exposé 17