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Curves with zero derivative in F-spaces

Published online by Cambridge University Press:  18 May 2009

N. J. Kalton
N. J. Kalton
Affiliation:
Department of Mathematics, University of Missouri-Columbia, Columbia Missouri 65211, U.S.A.
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Let X be an F-space (complete metric linear space) and suppose g:[0, 1] → X is a continuous map. Suppose that g has zero derivative on [0, 1], i.e.

for 0≤t≤1 (we take the left and right derivatives at the end points). Then, if X is locally convex or even if it merely possesses a separating family of continuous linear functionals, we can conclude that g is constant by using the Mean Value Theorem. If however X* = {0} then it may happen that g is not constant; for example, let X = Lp(0, 1) (0≤p≤1) and g(t) = l[0,t] (0≤t≤1) (the characteristic function of [0, t]). This example is due to Rolewicz [6], [7; p. 116].

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 22 , Issue 1 , January 1981 , pp. 19 - 29
Copyright
Copyright © Glasgow Mathematical Journal Trust 1981

References

REFERENCES

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