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On the representation of strictly continuous linear functionals

Published online by Cambridge University Press:  20 January 2009

Liaqat Ali Khan  and
K. Rowlands
Liaqat Ali Khan
Affiliation:
Department of MathematicsFederal Government CollegeIslamabadPakistan
K. Rowlands
Affiliation:
Department of Pure MathematicsThe University College of WalesAberystwyth
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Let X be a topological space, E a real or complex topological vector space, and C(X, E) the vector space of all bounded continuous E-valued functions on X; when E is the real or complex field this space will be denoted by C(X). The notion of the strict topology on C(X, E) was first introduced by Buck (1) in 1958 in the case of X locally compact and E a locally convex space. In recent years a large number of papers have appeared in the literature concerned with extending the results contained in Buck's paper. In particular, a number of these have considered the problem of characterising the strictly continuous linear functional on C(X, E); see, for example, (2), (3), (4) and (8). In this paper we suppose that X is a completely regular Hausdorff space and that E is a Hausdorff topological vector space with a non-trivial dual E′. The main result established is Theorem 3.2, where we prove a representation theorem for the strictly continuous linear functionals on the subspace Ctb(X, E) which consists of those functions f in C(X, E) such that f(X) is totally bounded.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 24 , Issue 2 , June 1981 , pp. 123 - 130
Copyright
Copyright © Edinburgh Mathematical Society 1981

References

REFERENCES

(1)Buck, R. C., Bounded continuous functions on a locally compact space, Michigan Math. J. 5 (1958), 95–104.CrossRefGoogle Scholar
(2)Fontenot, R. A., Strict topologies for vector-valued functions, Canadian J. Math. 26 (1974), 841–853.CrossRefGoogle Scholar
(3)Giles, R., A generalization of the strict topology, Trans. Amer. Math. Soc. 161 (1971), 467–474.Google Scholar
(4)Katsaras, A., Spaces of vector measures, Trans. Amer. Math. Soc. 206 (1975), 313–327.CrossRefGoogle Scholar
(5)Khan, L. A., The strict topology on a space of vector-valued functions, Proc. Edinburgh Math. Soc. 22 (1979), 35–41.Google Scholar
(6)Klee, V., Shrinkable neighbourhoods in Hausdorff linear spaces, Math. Ann. 141 (1960), 281–5.Google Scholar
(7)Waelbroeck, L., Topological vector spaces and algebras (Springer-Verlag Lecture Notes in Mathematics 230, 1971).Google Scholar
(8)Wells, J., Bounded continuous vector-valued functions on a locally compact space, Michigan J. Math. 11 (1965), 119–126.Google Scholar