§4 — Fréchet spaces are quasi-complete

Let $\mathbb K$ be a real-or-complex scalar field (in practice $\mathbb R$ or $\mathbb C$), and let $X$ be a $\mathbb K$-vector space that is also a real vector space with compatible scalar multiplications, carrying a uniform structure for which addition and subtraction are uniformly continuous (so the uniformity is the canonical translation-invariant uniformity of the induced topology), with scalar multiplication $\mathbb K\times X\to X$ jointly continuous, and Hausdorff. Assume $X$ is Fréchet. Fréchet (the bundle's definition, applied to the given uniform structure): locally convex as a real topological vector space, the topology is metrizable, and the space is complete for its uniform structure (every Cauchy filter converges). Then $X$ is quasi-complete over $\mathbb K$. Quasi-complete over $\mathbb K$ (the bundle's definition): for every index type $I$ (in the same universe as the space) carrying a partial order in which any two elements have a common upper bound, and every family $(v_i)_{i\in I}$ whose set of values is von Neumann bounded over $\mathbb K$ and which is Cauchy (for every neighbourhood $N$ of $0$ there is $i_0$ with $v_i-v_j\in N$ for all $i,j\ge i_0$), there is a point $x$ with $v_i-x\in N$ eventually (for all $i\ge$ some $i_0$) for every neighbourhood $N$ of $0$.