Definition 6

— A locally convex space E is said to be quasi-complete if every closed and bounded subset of E is complete (for the uniform structure induced by that of E). A complete locally convex space is quasi-complete, but the converse is not always true. For example, if E is an infinite dimensional Hilbert space, or more generally, an infinite dimensional reflexive Banach space, then E with its weakened topology is quasi-complete but not complete (II, p. 51, prop. 9). A quasi-complete space is semi-complete, since every Cauchy sequence is contained in a closed and bounded subset (III, p. 3, corollary and prop. 1). In particular, a locally convex metrizable and quasi-complete space is complete. In a Hausdorff quasi-complete space, the closure and the closed convex balanced envelope of a precompact subset are compact; in fact, they are precompact (II, p. 25, prop. 3), and complete being closed and bounded (III, p. 3, prop. 2).