A topological vector space X for which every barrelled bornivorous set in the space is a neighbourhood of 0 is called a quasi-barrelled space, where a set is bornivorous if it absorbs all bounded subsets of X.
3.
*If X is a barrelled space, then its topology coincides with the strong topology \ beta ( X, X') on X and with the Mackey topology on X generated by the pairing ( X, X').
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Given a barrelled space " X " and a locally convex space " Y ", then any family of pointwise bounded continuous linear mappings from " X " to " Y " is equicontinuous ( even uniformly equicontinuous ).
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:A linear operator from a barrelled space " X " to a Fr�chet space " Y " is continuous if and only if its graph is closed in the space " X " & times; " Y " equipped with the product topology.
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