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Banach Spaces of Continuous and Lipschitz functions

Banach spaces of continuous and Lipschitz functions
(FWF I 4570)

Project Abstract for a General Audience

The main purpose of our project is to study spaces of so-called continuous and Lipschitz functions—special kind of mathematical spaces consisting of “regular” and “nice” functions—in the context of different mathematical fields like geometry, topology, analysis and logic.

Intuitively speaking, continuous functions are those functions between two given mathematical spaces whose graphs do not have “holes’’ or “sudden jumps’’. Lipschitz functions are a particular type of continuous functions, where the degree of the “slope” of the function is everywhere bounded by a fixed positive number, so the graph of the function seems to be regular and tame. The spaces on which we consider these continuous and Lipschitz functions are so-called metric spaces. The main motivation for studying these metric spaces is to to generalize the physical space in which we live to more general settings. Here both the structure of the space and the way of measuring distances can be different from our everyday experience. Thus, our project can be thought as devoted to study properties of “transformations” (functions) between two special generalizations of
our physical world. In the recent decades a geometrical theory of these spaces, called metric geometry, has been developed and appeared to be very fruitful and promising.

The continuous and Lipschitz functions we consider form so-called Banach spaces—a special type of vector spaces where in addition to adding functions and multiplying them with numbers, we are able to introduce distances between functions and measure them. Introducing different types of distances between functions causes that those Banach spaces have also different geometrical properties. Thus, in relation to our aim of studying properties of metric spaces on which functions live we also plan to study properties of the Banach spaces built over these functions. As we already mentioned, we plan to use techniques and methods from different areas of mathematics such as geometry, analysis or logic. Such an interdisciplinary approach seems to be completely innovative and thus we believe that our project will allow us to deepen our understanding of those Banach
spaces and thus the metric spaces over which they are built.

Project Members

Publications

  1. J. Kąkol, D. Sobota and L. Zdomskyy, On complementability of \(c_0\) in spaces \(C(K\times L)\), Proc. Amer. Math. Soc. (2022), DOI: https://doi.org/10.1090/proc/16262, arXiv:2206.03794.
  2. C. Bargetz, J. Kąkol, and D. Sobota, On complemented copies of the space in spaces , Math. Nachr. (2023), 1–13.https://doi.org/10.1002/mana.202300026, arxiv:2107.03211
  3. W. Marciszewski, D. Sobota and L. Zdomskyy, On sequences of finitely supported measures related to the Josefson--Nissenzweig theorem, Topol. Appl. (2023), Special Issue on the Occasion of TopoSym 2023 accepted, arXiv:2303.03809.
  4. P. Borodulin-Nadzieja and D. Sobota, There is a P-measure in the random model. Fund. Math. (2023),  DOI: 10.4064/fm277-3-2023 , arXiv:2204.11694.
  5. D. Sobota and L. Zdomskyy, Convergence of measures after adding a real. Arch. Math. Logic (2023). DOI: 10.1007/s00153-023-00888-0, arXiv:2110.04568.
  6. D. Sobota and L. Zdomskyy, Minimally generated Boolean algebras with the Nikodym property, accepted in Topol. Appl. 323 (2023), DOI: 10.1016/j.topol.2022.108298, arXiv:2105.12467
  7. W. Marciszewski and D. Sobota, The Josefson-Nissenzweig theorem and filters on \(\omega\). Arch. Math. Logic (2023), accepted, arXiv:2204.01557

Preprints

  1. C. Bargetz, A. Bartoš, W. Kubiś, F. Luggin: Homogeneous isosceles-free spaces. Preprint (arXiv:2305.03163)
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