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A base norm space whose cone is not 1-generating

Published online by Cambridge University Press:  18 May 2009

David Yost
David Yost
Affiliation:
Department of Pure Mathematics, La Trobe University, Bundoora, 3083 Victoria, Australia
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Let E be an ordered Banach space with closed positive cone C. A base for C is a convex subset K of C with the property that every non-zero element of C has a unique representation of the form λk with λ > 0 and k ∈ K. Let S be the absolutely convex hull of K. If the Minkowski functional of S coincides with the given norm on E, then E is called a base norm space. Then K is a closed face of the unit ball of E, and S contains the open unit ball of E. Base norm spaces were first defined by Ellis [5, p. 731], although the special case of dual Banach spaces had been studied earlier by Edwards [4].

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 25 , Issue 1 , January 1984 , pp. 35 - 36
Copyright
Copyright © Glasgow Mathematical Journal Trust 1984

References

REFERENCES

1.Alfsen, E. M., Compact convex sets and boundary integrals (Springer-Verlag, 1971).CrossRefGoogle Scholar
2.Alfsen, E. M. and Effros, E. G., Structure in real Banach spaces. Part I, Ann of Math. (2), 96 (1972), 98–128.CrossRefGoogle Scholar
3.Asimow, L. and Ellis, A. J., Convexity theory and its applications in functional analysis (Academic Press, 1980).Google Scholar
4.Edwards, D. A., On the homeomorphic embedding of a locally compact cone into a Banach dual space equipped with the vague topology, Proc. London Math. Soc. (3), 14 (1964), 399–414.CrossRefGoogle Scholar
5.Ellis, A. J., The duality of partially ordered normed linear spaces, J. London Math. Soc. 39 (1964), 730–744.CrossRefGoogle Scholar
6.Hirsberg, B., M-ideals in complex function spaces and algebras, Israel J. Math. 12 (1972), 133–146.CrossRefGoogle Scholar
7.Hoffman, K., Banach spaces of analytic functions (Prentice-Hall, 1962).Google Scholar
8.Wong, Y. C. and Ng, K. F., Partially ordered topological vector spaces (Clarendon Press, 1973).Google Scholar