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Bounded-weak topologies and completeness in vector spaces

Published online by Cambridge University Press:  24 October 2008

G. T. Roberts
G. T. Roberts
Affiliation:
King's CollegeCambridge
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Extract

Suppose that X is any vector space on which it is possible to recognize a class of sets of such a nature that it is natural to call them ‘bounded’. (Precise conditions for such a class of sets are given in § 2.) Let L be any vector space of linear functionals on X which map each ‘bounded’ set into a bounded set. We say that a filter is boundedly-weakly convergent if it is convergent in the weak topology of the linear system [X, L] and contains a ‘bounded’ set. If M is a vector space of boundedly-weakly continuous linear functionals on X which includes L, we say that a subset S of M is limited if 〈 , f〉 converges uniformly for f ε S whenever is a boundedly-weakly convergent filter.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 49 , Issue 2 , April 1953 , pp. 183 - 189
Copyright
Copyright © Cambridge Philosophical Society 1953

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References

REFERENCES

(1)Arens, R.Duke math. J. 14 (1947), 787–94.Google Scholar
(2)Bourgin, D. C.Amer. J. Math. 65 (1943), 637–59.CrossRefGoogle Scholar
(3)Dieudonné, J.Proc. Amer. math. Soc. 1 (1950), 54–9.CrossRefGoogle Scholar
(4)Dieudonné, J. and Schwartz, L.Ann. Inst. Fourier, 1 (1949), 61–101.CrossRefGoogle Scholar
(5)Grothendieck, A.C.R. Acad. Sci., Paris, 230 (1950), 605–6.Google Scholar
(6)Koethe, G.Math. Nachr. 4 (1950), 70–80.Google Scholar
(7)Mackey, G. W.Trans. Amer. math. Soc. 57 (1945), 155–207.CrossRefGoogle Scholar