Some Ramsey type theorems for normed and quasinormed spaces

C. Henson; Nigel Kalton; N. Peck; Ignác Tereščák; Pavol Zlatoš

Studia Mathematica (1997)

  • Volume: 124, Issue: 1, page 81-100
  • ISSN: 0039-3223

Abstract

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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.

How to cite

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Henson, C., et al. "Some Ramsey type theorems for normed and quasinormed spaces." Studia Mathematica 124.1 (1997): 81-100. <http://eudml.org/doc/216398>.

@article{Henson1997,
abstract = {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.},
author = {Henson, C., Kalton, Nigel, Peck, N., Tereščák, Ignác, Zlatoš, Pavol},
journal = {Studia Mathematica},
keywords = {normed space; Banach space; quasinormed and quasi-Banach space; p-norm; biorthogonal sequence; uniformly independent sequence; irreducible sequence; Ramsey's Theorem; nonstandard analysis; -separated; -independent; quasinormed spaces},
language = {eng},
number = {1},
pages = {81-100},
title = {Some Ramsey type theorems for normed and quasinormed spaces},
url = {http://eudml.org/doc/216398},
volume = {124},
year = {1997},
}

TY - JOUR
AU - Henson, C.
AU - Kalton, Nigel
AU - Peck, N.
AU - Tereščák, Ignác
AU - Zlatoš, Pavol
TI - Some Ramsey type theorems for normed and quasinormed spaces
JO - Studia Mathematica
PY - 1997
VL - 124
IS - 1
SP - 81
EP - 100
AB - 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.
LA - eng
KW - normed space; Banach space; quasinormed and quasi-Banach space; p-norm; biorthogonal sequence; uniformly independent sequence; irreducible sequence; Ramsey's Theorem; nonstandard analysis; -separated; -independent; quasinormed spaces
UR - http://eudml.org/doc/216398
ER -

References

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  2. [2] J. Diestel, Sequences and Series in Banach Spaces, Springer, New York, 1984. 
  3. [3] C. W. Henson and L. C. Moore, Jr., The nonstandard theory of topological vector spaces, Trans. Amer. Math. Soc. 172 (1972), 405-435. Zbl0254.46001
  4. [4] C. W. Henson and P. Zlatoš, Indiscernibles and dimensional compactness, Comment. Math. Univ. Carolin. 37 (1996), 199-203. Zbl0851.46052
  5. [5] N. J. Kalton, Sequences of random variables in L p for p < 1, J. Reine Angew. Math. 329 (1981), 204-214. Zbl0461.60020
  6. [6] N. J. Kalton, Convexity, type and the three space problem, Studia Math. 69 (1981), 247-287. Zbl0499.46001
  7. [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. [8] V. L. Klee, On the Borelian and projective type of linear subspaces, Math. Scand. 6 (1958), 189-199. Zbl0088.08502
  9. [9] V. D. Milman, Geometric theory of Banach spaces, I, Russian Math. Surveys 25 (3) (1970), 111-170. 
  10. [10] J. Náter, P. Pulmann and P. Zlatoš, Dimensional compactness in biequivalence vector spaces, Comment. Math. Univ. Carolin. 33 (1992), 681-688. Zbl0784.46064
  11. [11] S. Rolewicz, On a certain class of linear metric spaces, Bull. Acad. Polon. Sci. Cl. III 5 (1957), 471-473. Zbl0079.12602
  12. [12] M. Šmíd and P. Zlatoš, Biequivalence vector spaces in the alternative set theory, Comment. Math. Univ. Carolin. 32 (1991), 517-544. Zbl0756.03027
  13. [13] E. M. Stein and N. J. Weiss, On the convergence of Poisson integrals, Trans. Amer. Math. Soc. 140 (1969), 35-54. Zbl0182.10801

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