Proposition 2

Every locally convex space E which is a Baire space (GT, IX, § 5, No. 3) is barrelled.

Let T be a barrel in E; since T is absorbent, E is the union of closed sets nT (n integer > 0); since E is a Baire space, at least one of these sets contains an interior point, hence T itself has an interior point x. If $x \neq 0$, since $-x \in T$, and 0 is a point of the open segment with extremities x and $-x$, 0 is an interior point of the convex set T (II, p. 14, prop. 16). Therefore T is a neighbourhood of 0.