Abstract
As opposed to the linear theory of Banach spaces which deals with the description of Banch spaces in terms of their linear topological properties involving the use of linear subspaces, linear maps and their relatives, the nonlinear theory seeks to achieve the same goal using nonlinear objects/quantities attached to a Banach space. These latter objects include Lipschitz maps, bilinear/multilinear maps and polynomials with domains of definition being replaced by subsets of the given space. It is remarkable that the linear structure of a Banach spaces is determined to a large extent by Lipschitz maps and that in a number of interesting cases, the linear structure is also captured by the larger class of uniformly continuous mappings acting between Banach spaces. In the present work, this will be illustrated by a number of results, both old and new, involving the extendability of Lipschitz maps and certain phenomena associated with it.
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Acknowledgements
This work was initiated during the author’s visit at the Harish Chandra Institute, Allahabad during Jan.–Feb.2015. He would like to thank his hosts for their kind invitation and for the hospitality provided to him during the period of his stay at the institute. Thanks are also due to the CSIR, New Delhi for providing support under its Emeritus Scientist Scheme vide grant No. 21 (0969)/13/EMR-II
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Sofi, M.A. (2016). Nonlinear Aspects of Certain Linear Phenomena in Banach Spaces. In: Cushing, J., Saleem, M., Srivastava, H., Khan, M., Merajuddin, M. (eds) Applied Analysis in Biological and Physical Sciences. Springer Proceedings in Mathematics & Statistics, vol 186. Springer, New Delhi. https://doi.org/10.1007/978-81-322-3640-5_26
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DOI: https://doi.org/10.1007/978-81-322-3640-5_26
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