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A highly non-smooth norm on Hilbert space

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Abstract

We show that there exists a familyF of unit balls in the Hilbert spacel 2 such that ∪F is dense inl 2 but the complement of ∪F is large in the sense of measure. In an appendix, we present a considerable simplification of the proof due to Preiss. As a corollary, we prove that there is an equivalent normp onl 2 such that the set of points wherep is Fréchet differentiable is Aronszajn null. This disproves a conjecture of Borwein and Noll in a very strong sense.

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Authors and Affiliations

  1. Department of Applied Mathematics, Charles University, Malostranské nám. 25, 118 00, Praha 1, Czech Republic

    Jiří Matoušek

  2. Institut für Theoretische Informatik, ETH Zürich, Switzerland

    Jiří Matoušek

  3. Mathematical Institute, Czech Academy of Sciences, Žitná 25, 11567, Praha, Czech Republic

    Eva Matoušková

  4. Institut für Mathematik, Johannes Kepler Universität, A-4040, Linz, Austria

    Eva Matoušková

Authors
  1. Jiří Matoušek
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  2. Eva Matoušková
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Corresponding author

Correspondence to Jiří Matoušek.

Additional information

Research supported by ETH Zürich, by Czech Republic Grant GAČR 0194/1996 and by Charles University grants No. 193,194/1996.

This author was supported by J. Kepler Universität and by the Czech Academy of Sciences grant GAAV-A1019705.

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Matoušek, J., Matoušková, E. A highly non-smooth norm on Hilbert space. Isr. J. Math. 112, 1–27 (1999). https://doi.org/10.1007/BF02773476

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  • DOI: https://doi.org/10.1007/BF02773476

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