Residuality in the set of norm attaining operators between Banach spaces Jung, Mingu Martín Suárez, Miguel Rueda Zoca, Abraham This paper was partially written when the first author was visiting the University of Granada and he would like to acknowledge the hospitality that he received there. The authors would like to thank Antonio Avilés, Luis Carlos García-Lirola, Gilles Godefroy, Manuel Maestre, Warren Moors, Vicente Montesinos, and Rafael Payá for kindly answering several inquiries related to the topics of the paper. We also thank the anonymous referee for the careful reading of the manuscript and for providing a number of comments which have improved its final form. M. Jung was supported by NRF (NRF-2019R1A2C1003857), by POSTECH Basic Science Research Institute Grant (NRF-2021R1A6A1A10042944) and by a KIAS Individual Grant (MG086601) at Korea Institute for Advanced Study. M. Martín was supported by Project PGC2018-093794-B-I00/AEI/10.13039/501100011033 (MCIU/AEI/FEDER, UE), by Junta de Andalucía I+D+i grants P20_00255, A-FQM-484-UGR18, and FQM-185, and by “Maria de Maeztu” Excellence Unit IMAG, reference CEX2020-001105-M funded by MCIN/AEI/10.13039/501100011033. A. Rueda Zoca was supported by Projects MTM2017-86182-P (Government of Spain, AEI/FEDER, EU), PGC2018-093794-B-I00/AEI/10.13039/501100011033 (MCIU/AEI/FEDER, UE), by Fundación Séneca, ACyT Región de Murcia grant 20797/PI/18, by Junta de Andalucía Grant A-FQM-484-UGR18, and by Junta de Andalucía Grant FQM-0185. We study the relationship between the residuality of the set of norm attaining functionals on a Banach space and the residuality and the denseness of the set of norm attaining operators between Banach spaces. Our first main result says that if C is a bounded subset of a Banach space X which admit an LUR renorming satisfying that, for every Banach space Y, the operators T from X to Y for which the supremum of with is attained are dense, then the set of those functionals which strongly exposes C is dense in ⁎. This extends previous results by J. Bourgain and K.-S. Lau. The particular case in which C is the unit ball of X, in which we get that the norm of ⁎ is Fréchet differentiable at a dense subset, improves a result by J. Lindenstrauss and we even present an example showing that Lindenstrauss' result was not optimal. In the reverse direction, we obtain results for the density of the set of absolutely strongly exposing operators from X to Y by requiring that the set of strongly exposing functionals on X is dense and conditions on Y or ⁎ involving RNP and discreteness on the set of strongly exposed points of Y or ⁎. These results include examples in which even the denseness of norm attaining operators was unknown. We also show that the residuality of the set of norm attaining operators implies the denseness of the set of absolutely strongly exposing operators provided the domain space and the dual of the range space are separable, extending a recent result for functionals. Finally, our results find important applications to the classical theory of norm-attaining operators, to the theory of norm-attaining bilinear forms, to the geometry of the preduals of spaces of Lipschitz functions, and to the theory of strongly norm-attaining Lipschitz maps. In particular, we solve a proposed open problem showing that the unique predual of the space of Lipschitz functions from the Euclidean unit circle fails to have Lindenstrauss property A. 2022-11-23T09:56:45Z 2022-11-23T09:56:45Z 2022-03-08 journal article M. Jung et al. Journal of Functional Analysis 284 (2023) 109746 [https://doi.org/10.1016/j.jfa.2022.109746] https://hdl.handle.net/10481/78090 10.1016/j.jfa.2022.109746 eng http://creativecommons.org/licenses/by-nc-nd/4.0/ open access Attribution-NonCommercial-NoDerivatives 4.0 Internacional