Distortion of Lipschitz functions on 𝑐₀(Γ)

Hájek, Petr; Novotný, Matěj

doi:10.1090/proc/13945

Distortion of Lipschitz functions on $c_0(\Gamma )$
HTML articles powered by AMS MathViewer

by Petr Hájek and Matěj Novotný PDF

Proc. Amer. Math. Soc. 146 (2018), 2173-2180 Request permission

Abstract:

Let $\Gamma$ be an uncountable cardinal. We construct a real symmetric $1$-Lipschitz function on the unit sphere of $c_0(\Gamma )$ whose restriction to any nonseparable subspace is a distortion.

References

Yoav Benyamini and Joram Lindenstrauss, Geometric nonlinear functional analysis. Vol. 1, American Mathematical Society Colloquium Publications, vol. 48, American Mathematical Society, Providence, RI, 2000. MR 1727673, DOI 10.1090/coll/048
Daniel P. Giesy, On a convexity condition in normed linear spaces, Trans. Amer. Math. Soc. 125 (1966), 114–146. MR 205031, DOI 10.1090/S0002-9947-1966-0205031-7
W. T. Gowers, Lipschitz functions on classical spaces, European J. Combin. 13 (1992), no. 3, 141–151. MR 1164759, DOI 10.1016/0195-6698(92)90020-Z
W. T. Gowers and B. Maurey, The unconditional basic sequence problem, J. Amer. Math. Soc. 6 (1993), no. 4, 851–874. MR 1201238, DOI 10.1090/S0894-0347-1993-1201238-0
Antonio J. Guirao, Vicente Montesinos, and Václav Zizler, Open problems in the geometry and analysis of Banach spaces, Springer, [Cham], 2016. MR 3524558, DOI 10.1007/978-3-319-33572-8
Petr Hájek, Vicente Montesinos Santalucía, Jon Vanderwerff, and Václav Zizler, Biorthogonal systems in Banach spaces, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 26, Springer, New York, 2008. MR 2359536
Robert C. James, Uniformly non-square Banach spaces, Ann. of Math. (2) 80 (1964), 542–550. MR 173932, DOI 10.2307/1970663
Thomas Jech, Set theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. The third millennium edition, revised and expanded. MR 1940513
B. Maurey, Symmetric distortion in $l_2$, Geometric aspects of functional analysis (Israel, 1992–1994) Oper. Theory Adv. Appl., vol. 77, Birkhäuser, Basel, 1995, pp. 143–147. MR 1353457
V. D. Milman, Geometric theory of Banach spaces. II. Geometry of the unit ball, Uspehi Mat. Nauk 26 (1971), no. 6(162), 73–149 (Russian). MR 0420226
Edward Odell and Thomas Schlumprecht, The distortion problem, Acta Math. 173 (1994), no. 2, 259–281. MR 1301394, DOI 10.1007/BF02398436
Edward Odell and Th. Schlumprecht, Distortion and asymptotic structure, Handbook of the geometry of Banach spaces, Vol. 2, North-Holland, Amsterdam, 2003, pp. 1333–1360. MR 1999198, DOI 10.1016/S1874-5849(03)80038-4
Baltasar Rodríguez-Salinas, On the complemented subspaces of $c_0(I)$ and $l_p(I)$ for $1<p<\infty$, Atti Sem. Mat. Fis. Univ. Modena 42 (1994), no. 2, 399–402. MR 1310440
Haskell P. Rosenthal, On relatively disjoint families of measures, with some applications to Banach space theory, Studia Math. 37 (1970), 13–36. MR 270122, DOI 10.4064/sm-37-1-13-36
Thomas Schlumprecht, An arbitrarily distortable Banach space, Israel J. Math. 76 (1991), no. 1-2, 81–95. MR 1177333, DOI 10.1007/BF02782845