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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.
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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

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References

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