Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-09T16:05:34.191Z Has data issue: false hasContentIssue false
  • English
  • Français

Weak and norm sequential convergence in M(S)

Published online by Cambridge University Press:  09 April 2009

A. Tong  and
D. Wilken
A. Tong
Affiliation:
Department of Mathematics State University of New York at Albany U.S.A.
D. Wilken
Affiliation:
Department of Mathematics State University of New York at Albany U.S.A.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let S be a compact Hausdorff space; let C(S) be the algebra of all continuous complex valued functions on S; and let M(S) be the dual space of (S) (the space of all regular Borel measures on S). In [2] Grothendieck gave a description of weak sequential convergence in M(S) in terms of uniform convergence on sequences of disjoint open sets in S. In this note we give a condition on the carriers of measures to guarantee that weak zero convergent sequences are norm zero convergent. While this condition is interesting in its own right, it can also be used to obtain immediately some well-known results about compact operators from C(S) to c0.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 15 , Issue 1 , February 1973 , pp. 1 - 6
Copyright
Copyright © Australian Mathematical Society 1973

References

[1]Dunford, N. and Schwartz, J., Linear Operators: Part I (Interscience, New York, 1958).Google Scholar
[2]Grothendieck, A., ‘Sur les applications linéaires faiblement compactes d'espaces du type C (K)’. Canadian Journal of Mathematics 5 (1953), 129–173.Google Scholar