Operator and dual operator bases in linear topological spaces
Author:
William B. Johnson
Journal:
Trans. Amer. Math. Soc. 166 (1972), 387400
MSC:
Primary 46A15
MathSciNet review:
0296643
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Abstract: A net of continuous linear projections of finite range on a Hausdorff linear topological space V is said to be a Schauder operator basisS.O.B. (resp. Schauder dual operator basisS.D.O.B.) provided it is pointwise bounded and converges pointwise to the identity operator on V, and (resp. ) whenever . S.O.B.'s and S.D.O.B.'s are natural generalizations of finite dimensional Schauder bases of subspaces. In fact, a sequence of operators is both a S.O.B. and S.D.O.B. iff it is the sequence of partial sum operators associated with a finite dimensional Schauder basis of subspaces. We show that many dualitytheory results concerning Schauder bases can be extended to S.O.B.'s or S.D.O.B.'s. In particular, a space with a S.D.O.B. is semireflexive if and only if the S.D.O.B. is shrinking and boundedly complete. Several results on S.O.B.'s and S.D.O.B.'s were previously unknown even in the case of Schauder bases. For example, Corollary IV.2 implies that the strong dual of an evaluable space which admits a shrinking Schauder basis is a complete barrelled space.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197202966436
PII:
S 00029947(1972)02966436
Keywords:
Schauder basis,
continuous linear projections,
duality
Article copyright:
© Copyright 1972
American Mathematical Society
