Abstract
The existence of a Banach limit as a translation invariant positive continuous linear functional on the space of bounded scalar sequences which is equal to 1 at the constant sequence \((1,1,\ldots ,1,\ldots )\) is proved in a first course on functional analysis as a consequence of the Hahn Banach extension theorem. Whereas its use as an important tool in classical summability theory together with its application in the existence of certain invariant measures on compact (metric) spaces is well known, a renewed interest in the theory of Banach limits has led to certain applications which have opened new vistas in the structure of Banach spaces. The paper is devoted to a discussion of certain developments, both classical and recent, surrounding the theory of Banach limits including the structure of the set of Banach limits with special emphasis on certain aspects of their applications to the existence of certain invariant measures, vector valued analogues of Banach limits, functional equations and in the structure theory of Banach spaces involving the existence of selectors of certain multi-valued mappings into the metric space of non-empty, convex, closed and bounded subsets of a Banach space with respect to the Hausdorff metric. The paper shall conclude with a brief description of some recent results of the author on the study of ‘simultaneous continuous linear’ operators (linear selections) involving Hahn Banach extensions on spaces of Lipschitz functions on (subspaces of) Banach spaces. Some open problems that naturally arise in this circle of ideas shall also be included at their appropriate places.
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Acknowledgements
Part of this work was carried out at the Harish Chandra Research Institute, Allahabad while the author was visiting the institute during Dec. 2018–Feb. 2019. He wishes to record his thanks to his host Prof. Ratnakumar for his kind invitation and hospitality during the author’s stay at the institute. Thanks are also due to the referee for careful reading of the manuscript and for his suggestions.
Funding
M. A. Sofi's work was funded by Council of Scientific and Industrial Research, India (No. 21(969)/13/EMR-II).
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Sofi, M.A. Banach limits: some new thoughts and perspectives. J Anal 29, 591–606 (2021). https://doi.org/10.1007/s41478-019-00184-2
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DOI: https://doi.org/10.1007/s41478-019-00184-2