Abstract
This article closes the cycle of characterizations of greedy-like bases in the “isometric” case initiated in Albiac and Wojtaszczyk (J. Approx. Theory 138(1):65–86, 2006) with the characterization of 1-greedy bases and continued in Albiac and Ansorena (J. Approx. Theory 201:7–12, 2016) with the characterization of 1-quasi-greedy bases. Here we settle the problem of providing a characterization of 1-almost greedy bases in Banach spaces. We show that a basis in a Banach space is almost greedy with almost greedy constant equal to 1 if and only if it has Property (A). This fact permits now to state that a basis is 1-greedy if and only if it is 1-almost greedy and 1-quasi-greedy. As a by-product of our work we also provide a tight estimate of the almost greedy constant of a basis in the non-isometric case.
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Acknowledgments
The authors feel indebted to the anonymous referee of an earlier version of this paper for pointing out to us how to shorten the proof of the Main Theorem.
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F. Albiac acknowledges the support of the Spanish Ministry for Economy and Competitivity Grants MTM2012-31286 for Operators, lattices, and structure of Banach spaces, and MTM2014-53009-P for Análisis Vectorial, Multilineal, y Aplicaciones. J.L. Ansorena acknowledges the support of the Spanish Ministry for Economy and Competitivity Grant MTM2014-53009-P for Análisis Vectorial, Multilineal, y Aplicaciones.
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Albiac, F., Ansorena, J.L. Characterization of 1-almost greedy bases. Rev Mat Complut 30, 13–24 (2017). https://doi.org/10.1007/s13163-016-0204-3
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DOI: https://doi.org/10.1007/s13163-016-0204-3