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Curves in Banach spaces which allow a C 2 ‐parameterization
Author(s) -
Duda Jakub,
Zajíček Luděk
Publication year - 2011
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdq100
Subject(s) - banach space , characterization (materials science) , mathematics , bounded function , parametrization (atmospheric modeling) , norm (philosophy) , pure mathematics , derivative (finance) , space (punctuation) , mathematical analysis , discrete mathematics , physics , computer science , quantum mechanics , political science , law , optics , economics , radiative transfer , operating system , financial economics
We give a complete characterization of those f : [0, 1] → X (where X is a Banach space that has a Fréchet smooth norm) that allow an equivalent C 2 ‐parameterization. For X = ℝ, a characterization is well known. However, even in the case X = ℝ 2 , several quite new ideas are needed. Moreover, the very close case of parameterizations with a bounded second derivative is solved.