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Operators on the generalized entire sequences

Published online by Cambridge University Press:  24 October 2008

I. J. Maddox
I. J. Maddox
Affiliation:
Queen's University of Belfast
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Extract

Let p = (pk) and q = (qk) be any sequences of strictly positive numbers. Denote by c0(p) the set of all complex sequences x = (xk) such that . The Köthe–Toeplitz dual of c0(p) was determined in Theorem 6(4). In the case in which p∈l∞ (the space of all bounded sequences), c0(p) becomes a locally convex FK space under the paranorm where M = max (1, sup pk). It has been recently proved by Lascarides(3) that the following three properties are equivalent:

(a) p∈c0(i.e.pk → 0),

(b) c0(p) is perfect (in the sense of Köthe-Toeplitz duality),

(c) c0(p) has the Schur property (i.e. weak and strong sequential convergence are equivalent).

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 71 , Issue 3 , May 1972 , pp. 491 - 494
Copyright
Copyright © Cambridge Philosophical Society 1972

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References

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