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(LF)-spaces with absolute bases

Published online by Cambridge University Press:  24 October 2008

G. Bennett  and
J. B. Cooper
G. Bennett
Affiliation:
St John's College, Cambridge
J. B. Cooper
Affiliation:
Clare College, Cambridge
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Extract

Suppose E is a locally convex space over a field K which can be the real line or the complex plane. Then a basis for E is a sequence (xk) of elements of E such that, if x ∈ E, x can be expressed uniquely as

where ξk ∈K for each k. If this representation converges absolutely, i.e. if

for every continuous seminorm p on E, then (xk) is called an absolute basis for E. If the mappings x → ξk from E into K are continuous for each k, then (xk) is a Schauder basis for E. The purpose of this paper is to prove some results for (LF)-spaces with bases and to use them to extend some theorems due to Pietsch. We recall that an (F)-space is a complete metrizable locally convex space and an (LF)-space the inductive limit of a strictly increasing sequence of (F)-spaces (En, τn) such that τn+1|En = τn for all n.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 67 , Issue 2 , March 1970 , pp. 283 - 286
Copyright
Copyright © Cambridge Philosophical Society 1970

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References

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