Abstract
If the topology of a locally convex space is the same as the strong topology, the space is said to be barrelled. If the topology is the same as the Mackey topology, and if every linear functional which is bounded on the bounded subsets of the space is continuous, then the space is said to be bornological. Every (F)-space has these properties, and each of these properties entails a sequence of important consequences. The investigation of the properties of barrelled and of bornological spaces, which generalises the theory of (F)-spaces, and which goes back to Mackey and to Bourbaki, forms an important part of the general theory of locally convex spaces. These two classes of spaces are considered in depth in § 27 and § 28.
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© 1983 Springer-Verlag Berlin, Heidelberg
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Köthe, G. (1983). Some Special Classes of Locally Convex Spaces. In: Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften, vol 159. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-64988-2_6
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DOI: https://doi.org/10.1007/978-3-642-64988-2_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-64990-5
Online ISBN: 978-3-642-64988-2
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