On a Lipschitz invariant of normed spaces
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- by Yoav Benyamini
- Proc. Amer. Math. Soc. 110 (1990), 979-981
- DOI: https://doi.org/10.1090/S0002-9939-1990-1028040-6
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Abstract:
C. Bessaga introduced an invariant $\eta (X)$ for $\sigma$-compact normed linear spaces. He showed that $\eta (X) = \eta (Y)$ whenever $X$ and $Y$ are Lipschitz homeomorphic. In this note we construct two $\sigma$-compact normed spaces with $\eta (X) = \eta (Y)$ which are not Lipschitz homeomorphic. Moreover, there are no compact convex sets $K$ and $L$ generating $X$ and $Y$, respectively, which are Lipschitz homeomorphic. This answers two problems posed by Bessaga.References
- 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
- C. Bessaga, A Lipschitz invariant of normed linear spaces related to the entropy numbers, Rocky Mountain J. Math. 10 (1980), no. 1, 81–84. MR 573863, DOI 10.1216/RMJ-1980-10-1-81
- Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces, Lecture Notes in Mathematics, Vol. 338, Springer-Verlag, Berlin-New York, 1973. MR 0415253
- Piotr Mankiewicz, On the differentiability of Lipschitz mappings in Fréchet spaces, Studia Math. 45 (1973), 15–29. MR 331055, DOI 10.4064/sm-45-1-15-29
Bibliographic Information
- © Copyright 1990 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 110 (1990), 979-981
- MSC: Primary 46B20
- DOI: https://doi.org/10.1090/S0002-9939-1990-1028040-6
- MathSciNet review: 1028040