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The weak-star closure of the unit ball in a subspace

Published online by Cambridge University Press:  20 January 2009

G. J. O. Jameson
G. J. O. Jameson
Affiliation:
Department of MathematicsUniversity of Lancaster
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Let X be a (real or complex) Banach space, and let K be a linear subspace of its dual X*. Denote by K1 the unit ball in K. If K is not weak-star closed, then the Krein-Šmul'yan theorem says that K1 is not weak-star closed. What, however, is its weak-star closure? Inner and outer extimates were obtained in [3] for the special case where K is a hyperplane. In the present paper we generalise these estimates to arbitrary linear subspaces. For f to belong to w*(K1) it is sufficient to have |φ(f)| ≦ 1 −∥f∥ for all φ in K0 (the annihilator of K in X**) with dist (φ, X)≦1. It is necessary to have |φ(f)| ≦ 1 + ∥f∥ for all such φ. These estimates depend on the action of each φ in K0 separately, which will often make them hard to apply in practice; in both cases, we derive a second estimate, expressed only in terms of certain constants that describe the position of X and K0 in X**.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 25 , Issue 1 , February 1982 , pp. 87 - 95
Copyright
Copyright © Edinburgh Mathematical Society 1982

References

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