Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-23T19:37:40.891Z Has data issue: false hasContentIssue false
  • English
  • Français

The sequential stability index of a function space

Part of: Linear function spaces and their duals

Published online by Cambridge University Press:  26 February 2010

J. M. Anderson  and
J. E. Jayne
J. M. Anderson
Affiliation:
Department of Mathematics, University College London.
J. E. Jayne
Affiliation:
Department of Mathematics, University College London.
Get access

Extract

One of the substantial differences between real and complex analysis is the behaviour of pointwise sequential limits of functions. It is well known that, if f(z) is a bounded analytic function in D = {z∈ C: |z| < 1}, then there exists a sequence {pn(z): n = 1,2,…} of polynomials such that

(i) ‖Pn‖ ≤ ‖f‖ for all n = 1,2,3,…, and

(ii) for each z ∈ D, Pn(Z) → f(z) as n → ∞,

where we have used the notation

MSC classification

Secondary: 46E15: Banach spaces of continuous, differentiable or analytic functions
Type
Research Article
Information
Mathematika , Volume 20 , Issue 2 , December 1973 , pp. 210 - 213
Copyright
Copyright © University College London 1973

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Jayne, J. E., “Descriptive set theory in compact spaces”, Amer. Math. Soc. Notices, 17 (1970), 268.Google Scholar
2. Kuratowski, K., “Sur une généralisation de la notion d'homéomorphie”, Fund. Math., 22 (1934), 206–220.CrossRefGoogle Scholar
3. Kuratowski, K., Topology, Vol. 1 (Academic Press, New York, 1966).Google Scholar
4. Landau, E., Darstellung und Begründung einiger neuerer Ergebnisse der Funktionentheorie (Berlin, 1929).Google Scholar
5. Lebesgue, H., “Sur les fonctions représentables analytiquement”, Jour, de Math., 1 (1905), 139–216.Google Scholar
6. Pelczήski, A. and Semadeni, Z., “Spaces of continuous functions III. (The space C(X) for X without perfect subsets.)”, Studia Math., 18 (1959), 211–222.CrossRefGoogle Scholar
7. Saks, S. and Zygmund, A., Analytic functions (Warszawa-Wroclaw, 1952).Google Scholar
8. Sarason, D., “On the order of a simply connected domain”, Michigan Math. J., 15 (1968), 129–133.CrossRefGoogle Scholar