Completeness in the Mackey topology by norming subspaces

Abstract

We study the class of Banach spaces X such that the locally convex space (X, µ(X, Y )) is complete for every norming and norm-closed subspace Y ⊂ X * , where µ(X, Y ) denotes the Mackey topology on X associated to the dual pair X, Y . Such Banach spaces are called fully Mackey complete. We show that fully Mackey completeness is implied by Efremov's property (E) and, on the other hand, it prevents the existence of subspaces isomorphic to ℓ 1 (ω 1 ). This extends previous results by Guirao, Montesinos and Zizler [J. Math. Anal. Appl. 445 (2017), 944-952] and Bonet and Cascales [Bull. Aust. Math. Soc. 81 (2010), [409][410][411][412][413]. Further examples of Banach spaces which are not fully Mackey complete are exhibited, like C[0, ω 1 ] and the long James space J(ω 1 ). Finally, by assuming the Continuum Hypothesis, we construct a Banach space with w * -sequential dual unit ball which is not fully Mackey complete. A key role in our discussion is played by the (at least formally) smaller class of Banach spaces X such that (Y, w * ) has the Mazur property for every norming and norm-closed subspace Y ⊂ X * .2010 Mathematics Subject Classification. 46A50, 46B26. Key words and phrases. Mackey topology; completeness; norming subspace; Mazur property. A.J. Guirao was supported by projects MTM2017-83262-C2-1-P (AEI/FEDER, UE) and 19368/PI/14 (Fundación Séneca). G. Martínez-Cervantes and J. Rodríguez were supported by projects MTM2014-54182-P and MTM2017-86182-P (AEI/FEDER, UE) and 19275/PI/14 (Fundación Séneca).

On the range of a vector measure

“…Proof of Corollary 11. Any fully Mackey complete Banach space cannot contain isomorphic copies of ℓ 1 (ω 1 ), see [9,Corollary 4.3]. Therefore, X contains no isomorphic copy of ℓ 1 (p) which, under the assumption that p > ω 1 , implies that (B X * , w * ) is convex block compact, see [14, 3D].…”

“…Following [9], the Banach space X is called fully Mackey complete if (X, µ(X, Y )) is complete for any norm-closed subspace Y ⊆ X * which is norming for X. Every Banach space having Efremov's property (E) is fully Mackey complete (see [9]). Thus, the next result (obtained under the set theoretic assumption that "p > ω 1 ") generalizes Corollary 2.…”

“…In this paper we push a bit further Freniche's techniques to obtain generalizations of his results above. Our discussion involves the Mazur property and the completeness of the Mackey topology of dual pairs associated to norming subspaces; these topics have been studied recently in [4,9,10,11]. Recall that a locally convex space E is said to have the Mazur property if every sequentially continuous linear functional from E to R is continuous.…”

“…Under the Continuum Hypothesis there exist Banach spaces separating both properties (see [2]), but it is unknown what happens in general. The relevance of property (E) to our discussion stems from the fact that if X has property (E), then (r(Z), w * ) has the Mazur property (see [9,Corollary 3.4]) and so does (Z, σ(Z, X)). As a consequence:…”

On integration in banach spaces and total sets

“…When Γ is norm-closed, (X, µ(X, Γ)) is complete if (and only if) it is quasi-complete (see e.g. [3]). This completeness assumption was used by Kunze [25] to find conditions ensuring the Γ-integrability of a Γ-scalarly integrable function provided that Γ is norming and norm-closed.…”