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NORMAL FUNCTIONALS ON LIPSCHITZ SPACES ARE WEAK* CONTINUOUS

Part of: Normed linear spaces and Banach spaces; Banach lattices Linear function spaces and their duals

Published online by Cambridge University Press:  08 April 2021

Ramón J. Aliaga  and
Eva Pernecká Open the ORCID record for Eva Pernecká [Opens in a new window]
Ramón J. Aliaga
Affiliation:
Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, Camino de Vera S/N, 46022 Valencia, Spain ([email protected])
Eva Pernecká
Affiliation:
Faculty of Information Technology, Czech Technical University in Prague, Thákurova 9, 160 00, Prague 6, Czech Republic ([email protected])
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Abstract

Let $\mathrm {Lip}_0(M)$ be the space of Lipschitz functions on a complete metric space M that vanish at a base point. We prove that every normal functional in ${\mathrm {Lip}_0(M)}^*$ is weak* continuous; that is, in order to verify weak* continuity it suffices to do so for bounded monotone nets of Lipschitz functions. This solves a problem posed by N. Weaver. As an auxiliary result, we show that the series decomposition developed by N. J. Kalton for functionals in the predual of $\mathrm {Lip}_0(M)$ can be partially extended to ${\mathrm {Lip}_0(M)}^*$ .

Keywords

Lipschitz-free spaceLipschitz functionLipschitz spacenormal functional

MSC classification

Primary: 46B20: Geometry and structure of normed linear spaces
Secondary: 46E15: Banach spaces of continuous, differentiable or analytic functions
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 21 , Issue 6 , November 2022 , pp. 2093 - 2102
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

Albiac, F. and Kalton, N. J., Topics in Banach Space Theory , 2nd ed., Graduate Texts in Mathematics, Vol. 233 (Springer, Cham, Switzerland, 2016).Google Scholar
Aliaga, R. J., Pernecká, E., Petitjean, C. and Procházka, A., Supports in Lipschitz-free spaces and applications to extremal structure, J. Math. Anal. Appl., 489 (2020), 124128.CrossRefGoogle Scholar
Cascales, B., Chiclana, R., Garicía-Lirola, L. C., Martín, M. and Rueda Zoca, A., On strongly norm attaining Lipschitz maps, J. Funct. Anal., 277 (2019), 1677–1717.CrossRefGoogle Scholar
Cúth, M., Kalenda, O. F. K. and Kaplický, P., Finitely additive measures and complementability of Lipschitz-free spaces, Israel J. Math., 230 (2019), 409–442.CrossRefGoogle Scholar
Godefroy, G., A survey on Lipschitz-free Banach spaces, Comment. Math., 55 (2015), 89–118.Google Scholar
Godefroy, G. and Kalton, N. J., Lipschitz-free Banach spaces, Studia Math., 159 (2003), 121–141.CrossRefGoogle Scholar
Godefroy, G., Lancien, G. and Zizler, V., The non-linear geometry of Banach spaces after Nigel Kalton, Rocky Mountain J. Math., 44 (2014), 1529–1583.CrossRefGoogle Scholar
Kadets, V., Martín, M. and Soloviova, M., Norm attaining Lipschitz functionals, Banach J. Math. Anal., 10 (2016), 621–637.CrossRefGoogle Scholar
Kalton, N. J., Spaces of Lipschitz and Hölder functions and their applications, Collect. Math., 55 (2004), 171–217.Google Scholar
Sakai, S., ${C}^{\ast }$ -Algebras and ${W}^{\ast }$ -Algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 60 (Springer, Berlin, 1971).Google Scholar
Schaefer, H. H., Banach Lattices and Positive Operators , Die Grundlehren der mathematischen Wissenschaften, Band 215 (Springer, New York, 1974).Google Scholar
Weaver, N., Lipschitz Algebras (World Scientific, River Edge, NJ, 1999).CrossRefGoogle Scholar
Weaver, N., Lipschitz Algebras, 2nd ed. (World Scientific, River Edge, NJ, 2018).CrossRefGoogle Scholar
Weaver, N., On the unique predual problem for Lipschitz spaces, Math. Proc. Camb. Philos. Soc. (3), 165 (2018), 467–473.CrossRefGoogle Scholar
Wojtaszczyk, P., Banach Spaces for Analysts , Cambridge Studies in Advanced Mathematics, Vol. 25 (Cambridge University Press, Cambridge, UK, 1991).Google Scholar