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1997 | 124 | 1 | 81-100
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Some Ramsey type theorems for normed and quasinormed spaces

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We prove that every bounded, uniformly separated sequence in a normed space contains a "uniformly independent" subsequence (see definition); the constants involved do not depend on the sequence or the space. The finite version of this result is true for all quasinormed spaces. We give a counterexample to the infinite version in $L_p[0,1]$ for each 0 < p < 1. Some consequences for nonstandard topological vector spaces are derived.
Twórcy
  • Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, Illinois 61801, U.S.A., henson@math.uiuc.edu
  • Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, Illinois 61801, U.S.A., peck@math.uiuc.edu
  • Faculty of Mathematics and Physics, Comenius University, Mlynská Dolina, 84215 Bratislava, Slovakia, terescak@fmph.uniba.sk
  • Faculty of Mathematics and Physics, Comenius University, Mlynská Dolina, 84215 Bratislava, Slovakia, zlatos@fmph.uniba.sk
  • Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, Illinois 61801, U.S.A.
Bibliografia
  • [1] T. Aoki, Locally bounded linear topological spaces, Proc. Imp. Acad. Tokyo 18 (1942), 588-594.
  • [2] J. Diestel, Sequences and Series in Banach Spaces, Springer, New York, 1984.
  • [3] C. W. Henson and L. C. Moore, Jr., The nonstandard theory of topological vector spaces, Trans. Amer. Math. Soc. 172 (1972), 405-435.
  • [4] C. W. Henson and P. Zlatoš, Indiscernibles and dimensional compactness, Comment. Math. Univ. Carolin. 37 (1996), 199-203.
  • [5] N. J. Kalton, Sequences of random variables in $L_p$ for p < 1, J. Reine Angew. Math. 329 (1981), 204-214.
  • [6] N. J. Kalton, Convexity, type and the three space problem, Studia Math. 69 (1981), 247-287.
  • [7] N. J. Kalton, N. T. Peck and J. W. Roberts, An F-space Sampler, London Math. Soc. Lecture Note Ser. 89, Cambridge Univ. Press, Cambridge, 1984.
  • [8] V. L. Klee, On the Borelian and projective type of linear subspaces, Math. Scand. 6 (1958), 189-199.
  • [9] V. D. Milman, Geometric theory of Banach spaces, I, Russian Math. Surveys 25 (3) (1970), 111-170.
  • [10] J. Náter, P. Pulmann and P. Zlatoš, Dimensional compactness in biequivalence vector spaces, Comment. Math. Univ. Carolin. 33 (1992), 681-688.
  • [11] S. Rolewicz, On a certain class of linear metric spaces, Bull. Acad. Polon. Sci. Cl. III 5 (1957), 471-473.
  • [12] M. Šmíd and P. Zlatoš, Biequivalence vector spaces in the alternative set theory, Comment. Math. Univ. Carolin. 32 (1991), 517-544.
  • [13] E. M. Stein and N. J. Weiss, On the convergence of Poisson integrals, Trans. Amer. Math. Soc. 140 (1969), 35-54.
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