The Analysis of Warner Boundedness

Abstract

In answer to Jarchow's 1981 text, we recently characterized when Cc(X) is a df -space, finding along the way attractive analytic characterizations of when the Tychonov space X is pseudocompact. Analogues now reveal how exquisitely Warner boundedness lies between these two notions. To illustrate, X is pseudocompact, X is Warner bounded or Cc(X) is a df -space if and only if for each sequence (µn)n ⊂ Cc(X) there exists a sequence (εn)n ⊂ (0, 1] such that (εnµn)n is weakly bounded, is strongly bounded or is equicontinuous, respectively. Our characterizations and proofs add to and simplify Warner's.

Weak barrelledness for C(X) spaces

“…We solve the thirty-year-old problem of Buchwalter and Schmets, generating other papers [7][8][9] with similar gains which may be thought of as wedding presents. Our primary contribution is to arrange the (re)marriage.…”

“…In [7][8][9] we answer, many times over, Jarchow's 1981 call to characterize C c (X) spaces that are df -spaces, and show that Theorem 1. X is pseudocompact, X is Warner bounded, or C c (X) is a df -space if and only if for each sequence (λ n ) n of continuous linear forms on C c (X) there exists a sequence (ε n ) n ⊂ (0, 1] such that (ε n λ n ) n is pointwise bounded, is uniformly bounded on bounded subsets of C c (X), or is equicontinuous, respectively.…”

“…In [7] ( [8]) we prove that the Tychonov space X is pseudocompact (is Warner bounded) if and only if the weak dual (the strong dual) of C c (X) is docile (compare with the above). The TweddleSaxon extension [32, Theorem 2.6] of the Tweddle-Yeomans criterion for enlargements of E can be simply stated as requiring algebraic complements of E to be weakly or strongly feral.…”