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Absolute convexity in certain topological linear spaces

Published online by Cambridge University Press:  24 October 2008

I. J. Maddox  and
J. W. Roles
I. J. Maddox
Affiliation:
University of Lancaster
J. W. Roles
Affiliation:
University of Lancaster
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Extract

For r > 0 a non-empty subset U of a linear space is said to be absolutely r-convex if x, y ∈ U and |λ|r + |μ|r ≤ 1 together imply λx + μy∈ U, or, equivalently, xl, …, xn∈ U and

It is clear that if U is absolutely r-convex, then it is absolutely s-convex whenever s < r. A topological linear space is said to be r-convex if every neighbourhood of the origin θ contains an absolutely r-convex neighbourhood of the origin. For the case 0 < r ≤ 1, these concepts were introduced and discussed by Landsberg(2).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 66 , Issue 3 , November 1969 , pp. 541 - 545
Copyright
Copyright © Cambridge Philosophical Society 1969

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References

REFERENCES

(1)Kelley, J. L., Namioka, I. et al. Linear topological spaces (Van Nostrand; Princeton, 1963).CrossRefGoogle Scholar
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