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In this paper we present a general “gliding hump” condition that implies the barrelledness of a normed vector space. Several examples of subspaces of are shown to be barrelled using the theorem. The barrelledness of the space of Pettis integrable functions is also implied by the theorem (this was first shown in [3]).
In this note an internal property of a ring of sets, named the Nested Partition Property, is shown to imply the Nikodym Property. A wide range of examples are shown to have this property.
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