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Norming subspaces of Banach spaces


Authors: V. P. Fonf, S. Lajara, S. Troyanski and C. Zanco
Journal: Proc. Amer. Math. Soc. 147 (2019), 3039-3045
MSC (2010): Primary 46B20, 46B10; Secondary 46B15
DOI: https://doi.org/10.1090/proc/14442
Published electronically: March 26, 2019
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Abstract: We show that if $ X$ is a closed subspace of a Banach space $ E$ and $ Z$ is a closed subspace of $ E^*$ such that $ Z$ is norming for $ X$ and $ X$ is total over $ Z$ (as well as $ X$ is norming for $ Z$ and $ Z$ is total over $ X$), then $ X$ and the preannihilator of $ Z$ are complemented in $ E$ whenever $ Z$ is $ w^*$-closed or $ X$ is reflexive.


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Additional Information

V. P. Fonf
Affiliation: Department of Mathematics, Ben-Gurion University of the Negev, 84105 Beer-Sheva, Israel
Email: fonf@math.bgu.ac.il

S. Lajara
Affiliation: Departamento de Matemáticas, Universidad de Castilla-La Mancha, Escuela de Ingenieros Industriales, 02071 Albacete, Spain
Email: sebastian.lajara@uclm.es

S. Troyanski
Affiliation: Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100 Murcia, Spain; and Institute of Mathematics and Informatics, Bulgarian Academy of Science, bl. 8, acad. G. Bonchev str., 1113 Sofia, Bulgaria
Email: stroya@um.es

C. Zanco
Affiliation: Dipartimento di Matematica, Università degli Studi, Via C. Saldini, 50, 20133 Milano MI, Italy
Email: clemente.zanco@unimi.it

DOI: https://doi.org/10.1090/proc/14442
Keywords: Norming subspace, total subspace, reflexive subspace, $M$-bibasic system
Received by editor(s): April 24, 2018
Received by editor(s) in revised form: September 10, 2018, September 28, 2018, and October 15, 2018
Published electronically: March 26, 2019
Additional Notes: The first and fourth authors were partially supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) of Italy.
The second author was supported by MICINN projects MTM2014-54182-P and MTM2015-65825-P (Spain), by MTM2017-86182-P (AEI/FEDER, UE) and by the Fundación Séneca (Agencia de Ciencia y Tecnología de la Región de Murcia) under project 19275/PI/14.
The third author was supported by MICINN project MTM2014-54182-P (Spain), by MTM2017-86182-P (AEI/FEDER, UE), by the Fundación Séneca (Agencia de Ciencia y Tecnología de la Región de Murcia) under project 19275/PI/14 and by the Bulgarian National Scientific Fund. DFNI-I02/10, 2015.
Dedicated: Dedicated to the memory of Professor Joseph Diestel
Communicated by: Stephen Dilworth
Article copyright: © Copyright 2019 American Mathematical Society