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Remarks on completeness in spaces of linear operators

Published online by Cambridge University Press:  17 April 2009

W. Ricker
W. Ricker
Affiliation:
School of Mathematics and Physics, Macquarie University, North Ryde, 2113, N.S.W., Australia.
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Abstract

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Whereas a locally convex Hausdorff space X inherits any completeness properties that the space of continuous linear operators, L (X), in X, may have (for the topology of pointwise convergence in X), this is not so in the converse situation and is the problem discussed here. The barrelledness of X in its Mackey topology plays an important role: if L (X) is quasicomplete, then X is barrelled for its Mackey topology. Consequently, for Mackey spaces X is turns out that L (X) is quasicomplete if and only if X is quasicomplete and barrelled: this is false if sequential completeness is substituted for quasicompleteness. Furthermore, there exist non-barrelled spaces X for which X and L (X) are quasicomplete (sequentially complete). Hence, although barrelledness is a sufficient condition for completeness of L (X) in various senses, it is certainly not necessary.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 34 , Issue 1 , August 1986 , pp. 25 - 35
Copyright
Copyright © Australian Mathematical Society 1986

References

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