Abstract
The Banach space \(\mathcal {P}({}^2X)\) of 2-homogeneous polynomials on the Banach space X can be naturally embedded in the Banach space \({\mathrm{Lip}_0}(B_X)\) of real-valued Lipschitz functions on \(B_X\) that vanish at 0. We investigate whether \(\mathcal {P}({}^2X)\) is a complemented subspace of \({\mathrm{Lip}_0}(B_X)\). This line of research can be considered as a polynomial counterpart to a classical result by Joram Lindenstrauss, asserting that \(\mathcal {P}({}^1X)=X^*\) is complemented in \({\mathrm{Lip}_0}(B_X)\) for every Banach space X. Our main result asserts that \(\mathcal {P}({}^2X)\) is not complemented in \({\mathrm{Lip}_0}(B_X)\) for every Banach space X with non-trivial type.
Similar content being viewed by others
Notes
We don’t use the standard notation \(\omega \) for a modulus of continuity as it would conflict with \(\omega \in \mathcal {O}_n\).
References
Albiac, F., Ansorena, J.L., Cúth, M., Doucha, M.: Lipschitz algebras and Lipschitz-free spaces over unbounded metric spaces. Int. Math. Res. Not. IMRN rnab193 (2021). https://doi.org/10.1093/imrn/rnab193
Albiac, F., Kalton, N.: Topics in Banach space theory, Graduate Texts in Mathematics, 233. Springer, New York (2006)
Aliaga, R.J., Pernecká, E.: Supports and extreme points in Lipschitz-free spaces. Rev. Mat. Iberoam. 36, 2073–2089 (2020)
Ambrosio, L., Puglisi, D.: Linear extension operators between spaces of Lipschitz maps and optimal transport. J. Reine Angew. Math. 764, 1–21 (2020)
Arias, A., Farmer, J.D.: On the structure of tensor products of \(\ell _p\)-spaces. Pacific J. Math. 175, 13–37 (1996)
Aron, R.M.: An introduction to polynomials on Banach spaces. Extract. Math. 17, 303–329 (2002)
Aron, R.M., Berner, P.D.: A Hahn-Banach extension theorem for analytic mappings. Bull. Soc. Math. Fr. 106, 3–24 (1978)
Aron, R.M., García, D., Maestre, M.: On norm attaining polynomials. Publ. Res. Inst. Math. Sci. 39, 165–172 (2003)
Aron, R.M., Hájek, P.: Odd degree polynomials on real Banach spaces. Positivity 11, 143–153 (2007)
Aron, R.M., Schottenloher, M.: Compact holomorphic mappings on Banach spaces and the approximation property. J. Funct. Anal. 21, 7–30 (1976)
Basso, G.: Computation of maximal projection constants. J. Funct. Anal. 277, 3560–3585 (2019)
Benyamini, Y., Gordon, Y.: Random factorization of operators between Banach spaces. J. Anal. Math. 39, 45–74 (1981)
Benyamini, Y., Lindenstrauss, J.: Geometric nonlinear functional analysis. Vol. 1, American Mathematical Society Colloquium Publications, 48. American Mathematical Society, Providence, RI (2000)
Candido, L., Cúth, M., Doucha, M.: Isomorphisms between spaces of Lipschitz functions. J. Funct. Anal. 277, 2697–2727 (2019)
Candido, L., Kaufmann, P.L.: On the geometry of Banach spaces of the form \({{\rm Lip}_0}(C(K))\). Proc. Am. Math. Soc. 149, 3335–3345 (2021)
Chevalley, C.: Theory of Lie Groups. I, Princeton Mathematical Series, 8. Princeton University Press, Princeton (1946)
Díaz, J.C., Dineen, S.: Polynomials on stable spaces. Ark. Mat. 36, 87–96 (1998)
Diestel, J., Jarchow, H., Tonge, A.: Absolutely summing operators, Cambridge Studies in Advanced Mathematics, 43. Cambridge University Press, Cambridge (1995)
Dineen, S., Mujica, J.: Banach spaces of homogeneous polynomials without the approximation property. Czechoslovak Math. J. 65, 367–374 (2015)
Fabian, M., Habala, P., Hájek, P., Montesinos Santalucía, V., Zizler, V.: Banach space theory. The basis for linear and nonlinear analysis, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Springer, New York (2011)
Falco, J., Garcia, D., Jung, M., Maestre, M.: Group invariant separating polynomials on a Banach space. Publ. Mat. 66, 207–233 (2022)
Farmer, J., Johnson, W.B.: Polynomial Schur and polynomial Dunford-Pettis properties. Banach spaces (Mérida, 1992), 95–105, Contemp. Math., 144, Amer. Math. Soc., Providence, RI (1993)
Figiel, T., Tomczak-Jaegermann, N.: Projections onto Hilbertian subspaces of Banach spaces. Israel J. Math. 33, 155–171 (1979)
Floret, K.: Natural norms on symmetric tensor products of normed spaces. Proceedings of the Second International Workshop on Functional Analysis (Trier, 1997). Note Mat. 17, 153–188 (1997)
Foucart, S., Lasserre, J.B.: Determining projection constants of univariate polynomial spaces. J. Approx. Theory 235, 74–91 (2018)
Foucart, S., Skrzypek, L.: On maximal relative projection constants. J. Math. Anal. Appl. 447, 309–328 (2017)
Godefroy, G., Kalton, N.J.: Lipschitz-free Banach spaces. Stud. Math. 159, 121–141 (2003)
Godefroy, G.: A survey on Lipschitz-free Banach spaces. Comment. Math. 55, 89–118 (2015)
Godefroy, G., Ozawa, N.: Free Banach spaces and the approximation properties. Proc. Am. Math. Soc. 142, 1681–1687 (2014)
Görlich, E., Markett, C.: A lower bound for projection operators on \(L_1(-1,1)\). Ark. Mat. 24, 81–92 (1986)
Grünbaum, B.: Projection constants. Trans. Am. Math. Soc. 95, 451–465 (1960)
Hájek, P., Johanis, M.: Smooth Analysis in Banach spaces. De Gruyter, Berlin (2014)
Hewitt, E., Ross, K.A.: Abstract Harmonic Analysis. Vol. I. Second edition, Grundlehren der Mathematischen Wissenschaften, 115. Springer, Berlin (1979)
Isbell, J.R., Semadeni, Z.: Projection constants and spaces of continuous functions. Trans. Am. Math. Soc. 107, 38–48 (1963)
James, R.C.: Nonreflexive spaces of type \(2\). Israel J. Math. 30, 1–13 (1978)
Jiménez-Vargas, A.: The approximation property for spaces of Lipschitz functions with the bounded weak\(^*\) topology. Rev. Mat. Iberoam. 34, 637–654 (2018)
Kadets, M.I., Snobar, M.G.: Some functionals over a compact Minkovskii space. Math. Notes 10, 694–696 (1971)
Kaufmann, P.L.: Products of Lipschitz-free spaces and applications. Stud. Math. 226, 213–227 (2015)
König, H.: Spaces with large projection constants. Israel J. Math. 50, 181–188 (1985)
Korovkin, P.P.: Linear operators and approximation theory. Translated from the Russian ed. (1959) Russian Monographs and Texts on Advanced Mathematics and Physics, Vol. III Gordon and Breach Publishers, Inc., New York; Hindustan Publishing Corp. (India), Delhi (1960)
Lancien, G., Pernecká, E.: Approximation properties and Schauder decompositions in Lipschitz-free spaces. J. Funct. Anal. 264, 2323–2334 (2013)
Lewicki, G.: Asymptotic estimate of absolute projection constants. Univ. Iagel. Acta Math. 51, 51–60 (2013)
Lindenstrauss, J.: Extension of compact operators. Mem. Amer. Math. Soc. 48 (1964)
Lindenstrauss, J.: On nonlinear projections in Banach spaces. Michigan Math. J. 11, 263–287 (1964)
Lindenstrauss, J.: On complemented subspaces of \(m\). Israel J. Math. 5, 153–156 (1967)
Lindenstrauss, J., Tzafriri, L.: Classical Banach spaces. Lecture Notes in Mathematics, 338. Springer, Berlin (1973)
Lozinski, S.: On a class of linear operators. Dokl. Akad. Nauk SSSR 61, 193–196 (1948)
Milman, V.D., Schechtman, G.: Asymptotic theory of finite-dimensional normed spaces. Lecture Notes in Mathematics, 1200. Springer, Berlin (1986)
Odyniec, W., Lewicki, G.: Minimal projections in Banach spaces. Lecture Notes in Mathematics, 1449. Springer, Berlin (1990)
Pełczyński, A.: On weakly compact polynomial operators on B-spaces with Dunford-Pettis property. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 11, 371–378 (1963)
Pernecká, E., Smith, R.J.: The metric approximation property and Lipschitz-free spaces over subsets of \(\mathbb{R}^N\). J. Approx. Theory 199, 29–44 (2015)
Pisier, G.: Martingales with values in uniformly convex spaces. Israel J. Math. 20, 326–350 (1975)
Positsel’skii, E.D.: Projection constants of symmetric spaces. Math. Notes 15, 430–435 (1974)
Rieffel, M.A.: Lipschitz extension constants equal projection constants, Operator theory, operator algebras, and applications, 147–162, Contemp. Math., 414, Amer. Math. Soc., Providence, RI (2006)
Rudin, W.: Projections on invariant subspaces. Proc. Am. Math. Soc. 13, 429–432 (1962)
Ryan, R.A.: Dunford-Pettis properties. Bull. Acad. Polon. Sci. Se’r. Sci. Math. 27, 373–379 (1979)
Themistoclakis, W., Van Barel, M.: Uniform approximation on the sphere by least squares polynomials. Numer. Algorithms 81, 1089–1111 (2019)
Themistoclakis, W., Van Barel, M.: Optimal Lebesgue constants for least squares polynomial approximation on the (hyper)sphere. arXiv:1808.03530
Tomczak-Jaegermann, N.: Banach-Mazur distances and finite-dimensional operator ideals. Pitman Monographs and Surveys in Pure and Applied Mathematics, 38. Longman Scientific & Technical, Harlow (1989)
Weaver, N.: Lipschitz Algebras. World Scientific Publishing, Hackensack, NJ (2018)
Weaver, N.: On the unique predual problem for Lipschitz spaces. Math. Proc. Camb. Philos. Soc. 165, 467–473 (2018)
Wojtaszczyk, P.: Banach Spaces for Analysts. Cambridge Studies in Advanced Mathematics, 25. Cambridge University Press, Cambridge (1991)
Acknowledgements
We wish to thank the anonymous referee for carefully reading our manuscript and for the helpful report.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Research of P. Hájek was supported in part by OPVVV CAAS CZ.02.1.01/0.0/0.0/16_019/0000778. Research of T. Russo was supported by the GAČR project 20-22230L; RVO: 67985840 and by Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of Istituto Nazionale di Alta Matematica (INdAM), Italy.
Rights and permissions
Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Hájek, P., Russo, T. Projecting Lipschitz Functions Onto Spaces of Polynomials. Mediterr. J. Math. 19, 190 (2022). https://doi.org/10.1007/s00009-022-02075-6
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00009-022-02075-6
Keywords
- Banach spaces of Lipschitz functions
- polynomials
- complemented subspaces
- Euclidean spaces
- type and cotype