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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Uniform homeomorphisms between the unit balls in $L_ p$ and $l_ p$
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by Gun-Marie Lövblom PDF
Proc. Amer. Math. Soc. 123 (1995), 405-409 Request permission


Let $T:B({L_p}) \to B({l_p}),1 \leq p < 2$, be a uniform homeomorphism with modulus of continuity ${\delta _T}$. It is shown that for any $\gamma ,0 \leq \gamma < \frac {{2 - p}}{{2p}}$, there exists $K > 0$ and a sequence $\{ {\varepsilon _n}\}$ with ${\varepsilon _n} \to 0$ such that $\delta _T^{ - 1}({\delta _T}({\varepsilon _n})) \geq K{\varepsilon _n}|\log {\varepsilon _n}{|^\gamma }$ for all ${\varepsilon _n}$.
  • J. Bourgain, Remarks on the extension of Lipschitz maps defined on discrete sets and uniform homeomorphisms, Geometrical aspects of functional analysis (1985/86), Lecture Notes in Math., vol. 1267, Springer, Berlin, 1987, pp. 157–167. MR 907692, DOI 10.1007/BFb0078143
  • Per Enflo, On the nonexistence of uniform homeomorphisms between $L_{p}$-spaces, Ark. Mat. 8 (1969), 103–105. MR 271719, DOI 10.1007/BF02589549
  • Yoav Benyamini, The uniform classification of Banach spaces, Texas functional analysis seminar 1984–1985 (Austin, Tex.), Longhorn Notes, Univ. Texas Press, Austin, TX, 1985, pp. 15–38. MR 832247
  • Joram Lindenstrauss, On nonlinear projections in Banach spaces, Michigan Math. J. 11 (1964), 263–287. MR 167821
  • Gun-Marie Lövblom, Uniform homeomorphisms between unit balls in $L_p$-spaces, Math. Scand. 62 (1988), no. 2, 294–302. MR 964229, DOI 10.7146/math.scand.a-12219
  • S. Mazur, Une remarque sur l’homéomorphie des champs fonctionnels, Studia Math. 1 (1929).
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Additional Information
  • © Copyright 1995 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 123 (1995), 405-409
  • MSC: Primary 46B99; Secondary 46B25, 46E30
  • DOI:
  • MathSciNet review: 1227523