Extremal properties of the set of vector-valued Banach limits

Francisco Javier García-Pacheco

Open Mathematics (2015)

  • Volume: 13, Issue: 1, page 761-765
  • ISSN: 2391-5455

Abstract

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In this manuscript we find another class of real Banach spaces which admit vector-valued Banach limits different from the classes found in [6, 7]. We also characterize the separating subsets of ℓ∞(X). For this we first need to study when the space of almost convergent sequences is closed in the space of bounded sequences, which turns out to happen only when the underlying space is complete. Finally, a study on the extremal structure of the set of vector-valued Banach limits is conducted when the underlying normed space is a Hilbert space.We also reach the conclusion that the set of vector-valued Banach limits is not a convex component of BCL(ℓ∞(X),X), provided that X is a 1-injective Banach space satisfying that the underlying compact Hausdorff topological space has isolated points.

How to cite

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Francisco Javier García-Pacheco. "Extremal properties of the set of vector-valued Banach limits." Open Mathematics 13.1 (2015): 761-765. <http://eudml.org/doc/276011>.

@article{FranciscoJavierGarcía2015,
abstract = {In this manuscript we find another class of real Banach spaces which admit vector-valued Banach limits different from the classes found in [6, 7]. We also characterize the separating subsets of ℓ∞(X). For this we first need to study when the space of almost convergent sequences is closed in the space of bounded sequences, which turns out to happen only when the underlying space is complete. Finally, a study on the extremal structure of the set of vector-valued Banach limits is conducted when the underlying normed space is a Hilbert space.We also reach the conclusion that the set of vector-valued Banach limits is not a convex component of BCL(ℓ∞(X),X), provided that X is a 1-injective Banach space satisfying that the underlying compact Hausdorff topological space has isolated points.},
author = {Francisco Javier García-Pacheco},
journal = {Open Mathematics},
keywords = {Banach limit; Almost convergence; Group of Isometries; Extremal Structure; transitive Banach space; separable Banach space; rotund Banach space},
language = {eng},
number = {1},
pages = {761-765},
title = {Extremal properties of the set of vector-valued Banach limits},
url = {http://eudml.org/doc/276011},
volume = {13},
year = {2015},
}

TY - JOUR
AU - Francisco Javier García-Pacheco
TI - Extremal properties of the set of vector-valued Banach limits
JO - Open Mathematics
PY - 2015
VL - 13
IS - 1
SP - 761
EP - 765
AB - In this manuscript we find another class of real Banach spaces which admit vector-valued Banach limits different from the classes found in [6, 7]. We also characterize the separating subsets of ℓ∞(X). For this we first need to study when the space of almost convergent sequences is closed in the space of bounded sequences, which turns out to happen only when the underlying space is complete. Finally, a study on the extremal structure of the set of vector-valued Banach limits is conducted when the underlying normed space is a Hilbert space.We also reach the conclusion that the set of vector-valued Banach limits is not a convex component of BCL(ℓ∞(X),X), provided that X is a 1-injective Banach space satisfying that the underlying compact Hausdorff topological space has isolated points.
LA - eng
KW - Banach limit; Almost convergence; Group of Isometries; Extremal Structure; transitive Banach space; separable Banach space; rotund Banach space
UR - http://eudml.org/doc/276011
ER -

References

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