---
_id: '8528'
abstract:
- lang: eng
  text: "In the present paper, we give a definition of prevalent (\"metrically prevalent\"
    ) sets in nonlinear function\r\nspaces. A subset of a Euclidean space is said
    to be metrically prevalent if its complement has measure zero.\r\nThere is no
    natural way to generalize the definition of a set of measure zero in a finite-dimensional
    space\r\nto the infinite-dimensional case [6]. Therefore, it is necessary to give
    a special definition of a metrically\r\nprevalent set (set of full measure) in
    an infinite-dimensional space. There are various ways to do so. We\r\nsuggest
    one of the possible ways to define the class of metrically prevalent sets in the
    space of smooth maps\r\nof one smooth manifold into another. It is shown in this
    paper that the class of metrically prevalent sets\r\nhas natural properties; in
    particular, the intersection of finitely many metrically prevalent sets is metrically\r\nprevalent.
    The main result of the paper is a prevalent version of Thorn's transversality
    theorem.\r\nIt is common practice in singularity theory and the theory of dynamical
    systems to say that a property\r\nholds for \"almost every\" map (or flow) if
    it holds for a residual set, i.e., a set that contains a countable\r\nintersection
    of open dense sets in the corresponding function space. However, even in finite-dimensional\r\nspaces
    such a set can have arbitrarily small (say, zero) Lebesgue measure. We prove that
    Thorn's transversality theorem holds for an essentially \"thicker\" set than a
    residual set. It seems reasonable to revise from\r\nthe prevalent point of view
    the classical results of singularity theory and theory of dynamical systems,\r\nincluding
    the multijet transversality theorem, Mather's stability theorem, Kupka-Smale's
    theorem for dynamical systems, etc. We shall do this elsewhere. The notion of
    prevalence in linear Banach spaces was\r\nintroduced and investigated in [8].
    One of the possible ways to define a class of prevalent sets in the space\r\nof
    smooth maps of manifolds, which essentially differs from that presented in this
    paper, is given in [7].\r\nDefinitions of typicalness based on the Lebesgue measure
    in a finite-dimensional space were suggested\r\nby Kolmogorov [10] and Arnold
    [11]. These definitions were cited and discussed in [9]. Here we only point\r\nout
    that the finite-dimensional analog of Arnold's definition allows prevalent sets
    to have arbitrarily small\r\nmeasure, whereas the prevalent sets in the sense
    of the finite-dimensional analog of the definition given in\r\nthe present paper
    are necessarily of full measure. Our definition is a modification of that due
    to Arnold.\r\nI wish to thank Yu. S. Illyashenko for constant attention to this
    work and useful discussions and\r\nR. I. Bogdanov for help in the preparation
    of this paper. "
article_processing_charge: No
article_type: original
author:
- first_name: Vadim
  full_name: Kaloshin, Vadim
  id: FE553552-CDE8-11E9-B324-C0EBE5697425
  last_name: Kaloshin
  orcid: 0000-0002-6051-2628
citation:
  ama: Kaloshin V. Prevalence in the space of finitely smooth maps. <i>Functional
    Analysis and Its Applications</i>. 1997;31(2):95-99. doi:<a href="https://doi.org/10.1007/bf02466014">10.1007/bf02466014</a>
  apa: Kaloshin, V. (1997). Prevalence in the space of finitely smooth maps. <i>Functional
    Analysis and Its Applications</i>. Springer Nature. <a href="https://doi.org/10.1007/bf02466014">https://doi.org/10.1007/bf02466014</a>
  chicago: Kaloshin, Vadim. “Prevalence in the Space of Finitely Smooth Maps.” <i>Functional
    Analysis and Its Applications</i>. Springer Nature, 1997. <a href="https://doi.org/10.1007/bf02466014">https://doi.org/10.1007/bf02466014</a>.
  ieee: V. Kaloshin, “Prevalence in the space of finitely smooth maps,” <i>Functional
    Analysis and Its Applications</i>, vol. 31, no. 2. Springer Nature, pp. 95–99,
    1997.
  ista: Kaloshin V. 1997. Prevalence in the space of finitely smooth maps. Functional
    Analysis and Its Applications. 31(2), 95–99.
  mla: Kaloshin, Vadim. “Prevalence in the Space of Finitely Smooth Maps.” <i>Functional
    Analysis and Its Applications</i>, vol. 31, no. 2, Springer Nature, 1997, pp.
    95–99, doi:<a href="https://doi.org/10.1007/bf02466014">10.1007/bf02466014</a>.
  short: V. Kaloshin, Functional Analysis and Its Applications 31 (1997) 95–99.
date_created: 2020-09-18T10:50:54Z
date_published: 1997-03-30T00:00:00Z
date_updated: 2021-01-12T08:19:54Z
day: '30'
doi: 10.1007/bf02466014
extern: '1'
intvolume: '        31'
issue: '2'
keyword:
- Applied Mathematics
- Analysis
language:
- iso: eng
month: '03'
oa_version: None
page: 95-99
publication: Functional Analysis and Its Applications
publication_identifier:
  issn:
  - 0016-2663
  - 1573-8485
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
status: public
title: Prevalence in the space of finitely smooth maps
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 31
year: '1997'
...