Stephen A. Saxon

(LF)-spaces, quasi-baire spaces and the strongest locally convex topology

on E. coarser than r, (n = 1,2 . . . . ). If E = 0 E, and ~ is the finest locally convex n = l topology on E which induces on each E, a topology coarser than %, then (E, ~) is said to be the inductive limit of the sequence {(E,, z,)}.% iThe family {(E,, z,)}.% 1 is a definin9 sequence for the inductive limit space (E, z). Note that we are using a narrow definition of the notion of an inductive limit that befits our needs. If each (E,, r.) is a Fr6chet (Banach) space, E. ~ E. + 1 and z is Hausdorff, then the inductive limit (E, z) is said to be an (LF)-space [(LB)-space], and we write

Weight of precompact subsets and tightness

Pfister (1976) and Cascales and Orihuela (1986) proved that precompact sets in (DF)- and (LM)-spaces have countable weight, i.e., are metrizable. Improvements by Valdivia (1982), Cascales and Orihuela (1987), and Kakol and Saxon (preprint) have varying methods of proof. For these and other improvements a refined method of upper semi-continuous compact-valued maps applied to uniform spaces will suffice. At the same time, this method allows us to dramatically improve Kaplanskys theorem, that the weak topology of metrizable spaces has countable tightness, extending it to include all (LM)-spaces and all quasi-barrelled (DF)-spaces, both in the weak and original topologies. One key is showing that for a large class G including all (DF)- and (LM)-spaces, countable tightness of the weak topology of E in G is equivalent to realcompactness of the weak∗ topology of the dual of E.

Generating varieties of topological groups

Recently several papers on varieties of topological groups have appeared. In this note we investigate the question: if Ω is a class of topological groups, what topological groups are in the variety V (Ω) generated by Ω that is, what topological groups can be “manufactured” from Ω using repeatedly the operations of taking subgroups, quotient groups and arbitrary cartesian products? We seeka general theorem which will be useful for investigating V (Ω) for well-known classesΩ.

Barrelled spaces with(out) separable quotients

While the separable quotient problem is famously open for Banach spaces, in the broader context of barrelled spaces we give negative solutions. Obversely, the study of pseudocompact \(X\) and Warner bounded \(X\) allows us to expand Rosenthals positive solution for Banach spaces of the form \(C_{c}(X)\) to barrelled spaces of the same form, and see that strong duals of arbitrary \(C_{c}(X)\) spaces admit separable quotients. DOI: 10.1017/S0004972714000422

Non-Baire hyperplanes in non-separable Baire spaces

Abstract Given an infinite-dimensional Banach space E, one may ask: Does E have (1) a properly separable quotient, (2) a dense non-Baire hyperplane? [Every closed hyperplane in a Baire space is Baire.] The famous separable quotient problem (1) remains unsolved, and question (2) is question 13.1.1 of P. Perez Carreras and J. Bonet (“North-Holland Math. Stud.,” Vol. 131, North-Holland, Amsterdam, 1987 ). In 1966 Wilansky-Klee conjectured the answer to (2) is always “no”; J. Arias de Reyna denied the conjecture (Math. Ann. 249, 1980, 111–114 ), proving the answer to (2) is “yes” whenever the answer to (1) is “yes,” under assumption of c-additivity (c-A), a condition weaker than Martins Axiom. M. Valdivia extended the result to E a Baire (2nd category in itself) topological vector space (Collect. Math. 34, 1983, 287–296). In general, (2) is non-trivial only in case E is Baire with E′ ≠ E ∗ . An extension of Valdivias extension yields the complete answer for (2) in the general locally convex setting, affirming question 13.1.1 of Perez Carreras and Bonet (referenced above) in particular: A [not necessarily] Hausdorff locally convex space E has a dense non-Baire hyperplane if and only if [ dim (E′) = ∞ and ] E′ ≠ E ∗ . Even for E non-locally convex, ( dim (E′) = ∞ and E′ ≠ E ∗ ) suffices, but is not necessary, whereas E′ ≠ E ∗ is obviously always necessary in order that E have a dense non-Baire hyperplane. Whether E′ ≠ E ∗ suffices in the Hausdorff non-locally convex case, and whether the assumption of c-A can be omitted, we do not know. Contrastingly, any infinite-dimensional E is the union of a sequence of hyperplanes, and if E is Baire, one of the hyperplanes must be dense and Baire; and every hyperplane of E must be unordered Baire-like if E is.

Weak Barrelledness Versus P-spaces

With Ferrando and Kąkol: \(X\) is a P-space if and only if every bounded sequence in \(C_{p}\left( X\right) \) is relatively compact, if and only if every pointwise eventually zero sequence in \(C_{p}\left( X\right) \) is summable. From these two extreme characterizations, already implicitly known, flow the better known ones. Weak barrelledness (with Sanchez Ruiz), briefly previewed here, affords clear motivation and prompted an open question herewith answered: The closed absolutely convex hull of every bounded countable union of absolutely convex compact sets in \(C_{p}\left( X\right) \) is compact if and only if every bounded set in \(C_{p}\left( X\right) \) is relatively compact (if and only if \(X\) is discrete).

Reducing the classical multipliers ℓ ∞ , C 0 and bv 0

For R ∈ { bv 0 , c 0 , l ∞ } a multiplier of FK spaces, the classical sectional convergence theorems permit the reduction of R to any of its dense barrelled subspaces as a simple consequence of the Closed Graph Theorem. (Cf. the Bachelis/Rosenthal reduction of R = l ∞ to its dense barrelled subspace m 0 .) A natural modern setting permits the reduction of R to any of the larger class of dense βφ subspaces. Bennett and Kaltons FK setting remarkably reduced R = l ∞ to any of its dense subspaces. This extreme reduction also obtains in the modern βφ setting since, surprisingly, every dense subspace of l ∞ is a βφ subspace. Moreover, the standard results, including the Bennett/Kalton reduction, easily follow from their βφ versions and the Closed Graph Theorem. Our two supporting papers find relevant “Non-barrelled dense βφ subspaces” and study “Generalized sectional convergence and multipliers”. Here we specialize the βφ approach to ordinary, particularly unconditional, sectional convergence.

Quasidistinguished countable enlargements of normed spaces

If E is a Hausdorff locally convex space and M is an -dimensional subspace of the algebraic dual E * that is transverse to the continuous dual E ′, then, according to [7], the Mackey topology τ( E , E ′ + M ) is a countable enlargement (CE) of τ( E , E ′) [or of E ]. Much is still unknown as to when CEs preserve barrelledness (cf. [14]). E is quasidistinguished (QD) if each bounded subset of the completion E is contained in the completion of a bounded subset of E [12]. Clearly, each normed space is QD, and Tsirulnikov [12] asked if each CE of a normed space must be a QDCE, i.e., must preserve the QD property. Since CEs preserve metrizability (but not normability), her question was whether metrizable spaces so obtained must be QD, and was moderated by Amemiyas negative answer (cf. [5, p. 404]) to Grothendiecks query, who had asked if all metrizable spaces are QD, having proved the separable ones are [4].