Weak barrelledness for C(X) spaces

Abstract

Buchwalter and Schmets reconciled C c (X) and C p (X) spaces with most of the weak barrelledness conditions of 1973, but could not determine if ℵ 0 -barrelled ⇔ ∞ -barrelled for C c (X). The areas grew apart. Full reconciliation with the fourteen conditions adopted by Saxon and Sánchez Ruiz needs their 1997 characterization of Ruess' property (L), which allows us to reduce the C c (X) problem to its 1973 status and solve it by carefully translating the topology of Kunen (1980) andvan Mill (1982) to find the example that eluded Buchwalter and Schmets. The more tractable C p (X) readily partitions the conditions into just two equivalence classes, the same as for metrizable locally convex spaces, instead of the five required for C c (X) spaces. Our paper elicits others, soon to appear, that analytically characterize when the Tychonov space X is pseudocompact, or Warner bounded, or when C c (X) is a df -space (Jarchow's 1981 question).

Bounded sets structure of CpX and quasi‐(DF)‐spaces

“…Then, as is dense in ℝ for some , the Baire space ℝ belongs also to the class . Now the main theorem of [32] applies to deduce that is countable, a contradiction (since then would be metrizable implying the finite-dimensionality of ).…”

“…If an lcs E is covered by a family of bounded sets satisfying conditions (i) and (ii) [and (iii)] we shall say that E has a [ fundamental ] bounded resolution (see also for more details). Let us recall that every metrizable lcs E has a fundamental bounded resolution (see, for example, Corollary below), while E has a fundamental sequence of bounded sets if and only if E is normable.…”

Tightness and distinguished Fréchet spaces

“…It is known (e.g., see [13,Corollary 3.3]) that a C c (X) space is a dual metric space if and only if it is a df -space, and we will use the two terms interchangeably in the C c (X) context. However, the paper [16] answers a 30-year-old question by showing that there does exist a C c (X) space that is a df -space and not a (DF)-space. For the moment, we will consider spaces C c (κ), where κ is an infinite cardinal, among which there is no distinction between df -and (DF)-spaces.…”

“…In [13] and [16] we show that there exists X for which C c (X) is a df -space and not a (DF)-space, more than answering the Buchwalter-Schmets [3] question from 1973. Thus the following corollary has a nonvacuous hypothesis and provides a class of spaces having the desired Valdivialike conclusion, and yet having no overlap with the spaces covered in Example 10, since the latter are all (DF)-spaces (Proposition 10).…”

Pseudocompact spaces $X$ and $df$-spaces $C_{c}\left ( X\right ) $

“…Any such C c (X), as we shall see, must be ∞ -barrelled and not ℵ 0 -barrelled, answering Buchwalter and Schmets' implied question [1, p. 349, Remark 1], even older than Jarchow's. The example is key to a companion paper [5] that completely resolves the weak barrelledness picture for C c (X) spaces.…”