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On the Cones Associated with Biorthogonal Systems and Bases in Banach Spaces

Published online by Cambridge University Press:  20 November 2018

C. W. Mcarthur ,
Ivan Singer  and
Mark Levin
C. W. Mcarthur
Affiliation:
The Romanian Academy of Sciences, Bucharest, Rumania
Ivan Singer
Affiliation:
The Florida State University, Tallahassee, Florida
Mark Levin
Affiliation:
The Florida State University, Tallahassee, Florida
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1. Let E be a Banach space (by this we shall mean, for simplicity, a real Banach space) and (xn,fn) ({xn} ⊂ E, {fn} ⊂ E*) a biorthogonal system, such that {fn} is total on E (i.e. the relations x ∈ E,fn(x) = 0, n = 1, 2, …, imply x = 0). Then it is natural to consider the cone

1

which we shall call “the cone associated with the biorthogonal system (xn,fn)”. In particular, if {xn} is a basis of E and {fn} the sequence of coefficient functional associated with the basis {xn}, this cone is nothing else but

2

and we shall call it “the cone associated with the basis {xn}”.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 1206 - 1217
Copyright
Copyright © Canadian Mathematical Society 1969

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