Finite codimensional linear isometries on spaces of differentiable and Lipschitz functions

Hironao Koshimizu

Open Mathematics (2011)

  • Volume: 9, Issue: 1, page 139-146
  • ISSN: 2391-5455

Abstract

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We characterize finite codimensional linear isometries on two spaces, C (n)[0; 1] and Lip [0; 1], where C (n)[0; 1] is the Banach space of n-times continuously differentiable functions on [0; 1] and Lip [0; 1] is the Banach space of Lipschitz continuous functions on [0; 1]. We will see they are exactly surjective isometries. Also, we show that C (n)[0; 1] and Lip [0; 1] admit neither isometric shifts nor backward shifts.

How to cite

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Hironao Koshimizu. "Finite codimensional linear isometries on spaces of differentiable and Lipschitz functions." Open Mathematics 9.1 (2011): 139-146. <http://eudml.org/doc/269211>.

@article{HironaoKoshimizu2011,
abstract = {We characterize finite codimensional linear isometries on two spaces, C (n)[0; 1] and Lip [0; 1], where C (n)[0; 1] is the Banach space of n-times continuously differentiable functions on [0; 1] and Lip [0; 1] is the Banach space of Lipschitz continuous functions on [0; 1]. We will see they are exactly surjective isometries. Also, we show that C (n)[0; 1] and Lip [0; 1] admit neither isometric shifts nor backward shifts.},
author = {Hironao Koshimizu},
journal = {Open Mathematics},
keywords = {Linear isometry; Finite codimension; Continuously differentiable function; Lipschitz continuous function; The Banach-Stone theorem; Shift; finite codimensional isometries; spaces of differentiable functions; Banach-Stone theorem; shift},
language = {eng},
number = {1},
pages = {139-146},
title = {Finite codimensional linear isometries on spaces of differentiable and Lipschitz functions},
url = {http://eudml.org/doc/269211},
volume = {9},
year = {2011},
}

TY - JOUR
AU - Hironao Koshimizu
TI - Finite codimensional linear isometries on spaces of differentiable and Lipschitz functions
JO - Open Mathematics
PY - 2011
VL - 9
IS - 1
SP - 139
EP - 146
AB - We characterize finite codimensional linear isometries on two spaces, C (n)[0; 1] and Lip [0; 1], where C (n)[0; 1] is the Banach space of n-times continuously differentiable functions on [0; 1] and Lip [0; 1] is the Banach space of Lipschitz continuous functions on [0; 1]. We will see they are exactly surjective isometries. Also, we show that C (n)[0; 1] and Lip [0; 1] admit neither isometric shifts nor backward shifts.
LA - eng
KW - Linear isometry; Finite codimension; Continuously differentiable function; Lipschitz continuous function; The Banach-Stone theorem; Shift; finite codimensional isometries; spaces of differentiable functions; Banach-Stone theorem; shift
UR - http://eudml.org/doc/269211
ER -

References

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