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Research output: Contribution to journal › Comment/Letter to the editor › Scientific
The author regrets a mistake made in Kraaij . We summarize the results which remain valid and those whose validity is now unclear.
An overview of the status of the main results: Let X be a separable metric space. On X we consider the space of bounded continuous functions [Formula presented] equipped with the strict topology, cf. Sentilles . In addition, let [Formula presented] be the space of τ-additive Borel measures on X and let σ be the weak topology on [Formula presented] induced by [Formula presented]. In Kraaij , four additional topologies were considered on [Formula presented]: • σ lf, the finest locally convex topology on [Formula presented] that coincides with σ on all β-equicontinuous sets in [Formula presented],• σ f, the finest topology on [Formula presented] that coincides with σ on all β-equicontinuous sets in [Formula presented],• kσ the finest topology on [Formula presented] that coincides with σ on all weakly compact sets in [Formula presented],• β ∘ the polar topology on [Formula presented] generated using all pre-compact sets in [Formula presented], cf. Köthe .The main result of Kraaij  is Theorem 1.7 that states that σ lf = σ f = kσ = β ∘. The following result remains true Proposition 1.1 σ f = kσ and σ lf = β ∘. As a consequence of the missing identification σ f = σ lf, it is unclear whether [Formula presented] is infra-Pták by using Proposition 1.2. This in turn leads to the failure of establishing Corollaries 1.10, 1.11 and 1.12. Proposition 1.6 and Lemma's 1.8 and 1.9 are established using results in the literature and remain valid as it is. The mistake and an overview of its consequences in the proof sections: The proof that σ f = σ lf was based on the observation that as σ lf⊆σ f it suffices to verify that σ f is locally convex. This was carried out in two steps. Step 1: σ f was explicitly identified as a quotient topology T and it was shown that kσ = T = σ f. This part remains valid. Step 2: The explicit characterization T was then used to establish that T is locally convex. This part contains an error in the proof of Lemma 2.7. As a consequence, is unclear whether Lemma 2.8 and Proposition 2.6 remain true. The error: The proof that addition is a continuous map for T is mistaken, the proof that scalar multiplication is continuous remains valid. The exact mistake in Lemma 2.7 is the claim that H⊆U. Let [Formula presented] be the map defined by [Formula presented]. Let A, B be σ + open subsets in [Formula presented] for and let C be σ open in M τ. Finally let [Formula presented]. The sets H and U were defined as [Formula presented] The issue is that ⊕ adds the set C interpreted as the diagonal [Formula presented] to the set A×B, whereas the construction to obtain H adds the much larger product space C×C to [Formula presented]. As a consequence the claim H⊆U remains unproven.
Research output: Contribution to journal › Article › Scientific › peer-review