Proposition 4

Let E be a vector space and let $(E_i)_{i \in I}$ be a family of locally convex spaces. For each $i \in I$, let $f_i$ be a linear mapping of E in $E_i$; then the topology $\mathscr{T}$ on E, which is the coarsest making each mapping $f_i$ continuous, is a locally convex topology.

Using II, p. 24, corollary, this is a particular case of the corresponding property for topologies defined by semi-norms (II, p. 5).

In particular, every vector subspace of a locally convex space, and every product space of locally convex spaces, is locally convex. Every projective limit of locally convex spaces is locally convex.

Every enumerable product of Fréchet spaces (and in particular every enumerable product of Banach spaces) is a Fréchet space.

Every locally convex Hausdorff space E is isomorphic to a subspace of a product of Banach spaces and this subspace is closed if E is complete (II, p. 5, prop. 3). Every Fréchet space is isomorphic to a closed subspace of an enumerable product of Banach spaces (loc. cit.).