Proposition 9
Tag 07LO, permanent. Cite it as [Bourbaki, Tag 07LO].
Topological Vector Spaces, chapter III, § 1. Bornology in a topological vector space
— (i) A closed vector subspace of a quasi-complete locally convex space is quasi-complete. (ii) The product of quasi-complete locally convex spaces is quasi-complete. (iii) The topological direct sum of quasi-complete locally convex spaces is quasi-complete. (iv) A locally convex space which is the strict inductive limit of a sequence of closed quasi-complete subspaces is quasi-complete. Assertion (i) follows from Remark 2 (III, p. 2), (ii) from III, p. 4, cor. 2, (iii) from prop. 5 (III, p. 5) and (iv) from prop. 6 (III, p. 5).
We shall see later that the quotient space of a quasi-complete locally convex space by a closed vector space is not necessarily quasi-complete (IV, p. 63, exerc. 10).
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
Cited by
Nothing.