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Tytuł artykułu

PCA sets and convexity

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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].
Słowa kluczowe
  • Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, IL 61801, U.S.A.,
  • [1] H. Becker, Pointwise limits of sequences and $Σ^1_2$ sets, Fund. Math. 128 (1987), 159-170.
  • [2] H. Becker, S. Kahane and A. Louveau, Some complete $Σ^1_2$ sets in harmonic analysis, Trans. Amer. Math. Soc. 339 (1993), 323-336.
  • [3] B. Bossard, Théorie descriptive des ensembles en géométrie des espaces de Banach, thèse, Univ. Paris VII, 199?.
  • [4] B. Bossard, Co-analytic families of norms on a separable Banach space, Illinois J. Math. 40 (1996), 162-181.
  • [5] B. Bossard, G. Godefroy and R. Kaufman, Hurewicz's theorems and renorming of Banach spaces, J. Funct. Anal. 140 (1996), 142-150.
  • [6] R. G. Bourgin, Geometric Aspects of Convex Sets with Radon-Nikodým Property, Lecture Notes in Math. 993, Springer, 1983.
  • [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.
  • [8] G. A. Edgar, A noncompact Choquet theorem, Proc. Amer. Math. Soc. 49 (1975), 354-358.
  • [9] J. E. Jayne and C. A. Rogers, The extremal structure of convex sets, J. Funct. Anal. 26 (1977), 251-288.
  • [10] R. Kaufman, Co-analytic sets and extreme points, Bull. London Math. Soc. 19 (1987), 72-74.
  • [11] R. Kaufman, Topics on analytic sets, Fund. Math. 139 (1991), 217-229.
  • [12] R. Kaufman, Extreme points and descriptive sets, ibid. 143 (1993), 179-181.
  • [13] R. R. Phelps, Lectures on Choquet's Theorem, Van Nostrand Math. Stud. 7, Van Nostrand, Princeton, NJ, 1966.
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