PCA sets and convexity

R. Kaufman

Fundamenta Mathematicae (2000)

  • Volume: 163, Issue: 3, page 267-275
  • ISSN: 0016-2736

Abstract

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Three sets occurring in functional analysis are shown to be of class PCA (also called Σ 2 1 ) and to be exactly of that class. The definition of each set is close to the usual objects of modern analysis, but some subtlety causes the sets to have a greater complexity than expected. Recent work in a similar direction is in [1, 2, 10, 11, 12].

How to cite

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Kaufman, R.. "PCA sets and convexity." Fundamenta Mathematicae 163.3 (2000): 267-275. <http://eudml.org/doc/212443>.

@article{Kaufman2000,
abstract = {Three sets occurring in functional analysis are shown to be of class PCA (also called $Σ^1_2$) and to be exactly of that class. The definition of each set is close to the usual objects of modern analysis, but some subtlety causes the sets to have a greater complexity than expected. Recent work in a similar direction is in [1, 2, 10, 11, 12].},
author = {Kaufman, R.},
journal = {Fundamenta Mathematicae},
keywords = {norm; Banach space; convex sets; integrals over extreme points; Borel set; PCA set; analytic set; co-analytic set},
language = {eng},
number = {3},
pages = {267-275},
title = {PCA sets and convexity},
url = {http://eudml.org/doc/212443},
volume = {163},
year = {2000},
}

TY - JOUR
AU - Kaufman, R.
TI - PCA sets and convexity
JO - Fundamenta Mathematicae
PY - 2000
VL - 163
IS - 3
SP - 267
EP - 275
AB - Three sets occurring in functional analysis are shown to be of class PCA (also called $Σ^1_2$) and to be exactly of that class. The definition of each set is close to the usual objects of modern analysis, but some subtlety causes the sets to have a greater complexity than expected. Recent work in a similar direction is in [1, 2, 10, 11, 12].
LA - eng
KW - norm; Banach space; convex sets; integrals over extreme points; Borel set; PCA set; analytic set; co-analytic set
UR - http://eudml.org/doc/212443
ER -

References

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  1. [1] H. Becker, Pointwise limits of sequences and Σ 2 1 sets, Fund. Math. 128 (1987), 159-170. 
  2. [2] H. Becker, S. Kahane and A. Louveau, Some complete Σ 2 1 sets in harmonic analysis, Trans. Amer. Math. Soc. 339 (1993), 323-336. Zbl0799.04006
  3. [3] B. Bossard, Théorie descriptive des ensembles en géométrie des espaces de Banach, thèse, Univ. Paris VII, 199?. 
  4. [4] B. Bossard, Co-analytic families of norms on a separable Banach space, Illinois J. Math. 40 (1996), 162-181. 
  5. [5] B. Bossard, G. Godefroy and R. Kaufman, Hurewicz's theorems and renorming of Banach spaces, J. Funct. Anal. 140 (1996), 142-150. Zbl0872.46003
  6. [6] R. G. Bourgin, Geometric Aspects of Convex Sets with Radon-Nikodým Property, Lecture Notes in Math. 993, Springer, 1983. Zbl0512.46017
  7. [7] R. Deville, G. Godefroy and V. Zizler, Smoothness and Renormings in Banach Spaces, Pitman Monogr. Surveys Pure Appl. Math. 64, Longman Sci. Tech., 1993. Zbl0782.46019
  8. [8] G. A. Edgar, A noncompact Choquet theorem, Proc. Amer. Math. Soc. 49 (1975), 354-358. Zbl0273.46012
  9. [9] J. E. Jayne and C. A. Rogers, The extremal structure of convex sets, J. Funct. Anal. 26 (1977), 251-288. Zbl0411.46008
  10. [10] R. Kaufman, Co-analytic sets and extreme points, Bull. London Math. Soc. 19 (1987), 72-74. Zbl0601.54041
  11. [11] R. Kaufman, Topics on analytic sets, Fund. Math. 139 (1991), 217-229. Zbl0764.28002
  12. [12] R. Kaufman, Extreme points and descriptive sets, ibid. 143 (1993), 179-181. Zbl0832.54031
  13. [13] R. R. Phelps, Lectures on Choquet's Theorem, Van Nostrand Math. Stud. 7, Van Nostrand, Princeton, NJ, 1966. Zbl0135.36203

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