Bessaga's conjecture in unstable Köthe spaces and products

Zefer Nurlu; Jasser Sarsour

Studia Mathematica (1993)

  • Volume: 104, Issue: 3, page 221-228
  • ISSN: 0039-3223

Abstract

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Let F be a complemented subspace of a nuclear Fréchet space E. If E and F both have (absolute) bases ( e n ) resp. ( f n ) , then Bessaga conjectured (see [2] and for a more general form, also [8]) that there exists an isomorphism of F into E mapping f n to t n e π ( k n ) where ( t n ) is a scalar sequence, π is a permutation of ℕ and ( k n ) is a subsequence of ℕ. We prove that the conjecture holds if E is unstable, i.e. for some base of decreasing zero-neighborhoods ( U n ) consisting of absolutely convex sets one has ∃s ∀p ∃q ∀r l i m n ( d n + 1 ( U q , U p ) ) / ( d n ( U r , U s ) ) = 0 where d n ( U , V ) denotes the nth Kolmogorov diameter.

How to cite

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Nurlu, Zefer, and Sarsour, Jasser. "Bessaga's conjecture in unstable Köthe spaces and products." Studia Mathematica 104.3 (1993): 221-228. <http://eudml.org/doc/215971>.

@article{Nurlu1993,
abstract = {Let F be a complemented subspace of a nuclear Fréchet space E. If E and F both have (absolute) bases $(e_n)$ resp. $(f_n)$, then Bessaga conjectured (see [2] and for a more general form, also [8]) that there exists an isomorphism of F into E mapping $f_n$ to $t_n e_\{π(k_n)\}$ where $(t_n)$ is a scalar sequence, π is a permutation of ℕ and $(k_n)$ is a subsequence of ℕ. We prove that the conjecture holds if E is unstable, i.e. for some base of decreasing zero-neighborhoods $(U_n)$ consisting of absolutely convex sets one has ∃s ∀p ∃q ∀r $lim_n (d_\{n+1\}(U_q, U_p))/(d_n(U_r, U_s)) = 0$ where $d_n(U,V)$ denotes the nth Kolmogorov diameter.},
author = {Nurlu, Zefer, Sarsour, Jasser},
journal = {Studia Mathematica},
keywords = {complemented subspace of a nuclear Fréchet space; Kolmogorov diameter},
language = {eng},
number = {3},
pages = {221-228},
title = {Bessaga's conjecture in unstable Köthe spaces and products},
url = {http://eudml.org/doc/215971},
volume = {104},
year = {1993},
}

TY - JOUR
AU - Nurlu, Zefer
AU - Sarsour, Jasser
TI - Bessaga's conjecture in unstable Köthe spaces and products
JO - Studia Mathematica
PY - 1993
VL - 104
IS - 3
SP - 221
EP - 228
AB - Let F be a complemented subspace of a nuclear Fréchet space E. If E and F both have (absolute) bases $(e_n)$ resp. $(f_n)$, then Bessaga conjectured (see [2] and for a more general form, also [8]) that there exists an isomorphism of F into E mapping $f_n$ to $t_n e_{π(k_n)}$ where $(t_n)$ is a scalar sequence, π is a permutation of ℕ and $(k_n)$ is a subsequence of ℕ. We prove that the conjecture holds if E is unstable, i.e. for some base of decreasing zero-neighborhoods $(U_n)$ consisting of absolutely convex sets one has ∃s ∀p ∃q ∀r $lim_n (d_{n+1}(U_q, U_p))/(d_n(U_r, U_s)) = 0$ where $d_n(U,V)$ denotes the nth Kolmogorov diameter.
LA - eng
KW - complemented subspace of a nuclear Fréchet space; Kolmogorov diameter
UR - http://eudml.org/doc/215971
ER -

References

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