On monotonic functions from the unit interval into a Banach space with uncountable sets of points of discontinuity

Artur Michalak. "On monotonic functions from the unit interval into a Banach space with uncountable sets of points of discontinuity." Studia Mathematica 155.2 (2003): 171-182. <http://eudml.org/doc/286607>.

@article{ArturMichalak2003,
abstract = {We say that a function f from [0,1] to a Banach space X is increasing with respect to E ⊂ X* if x* ∘ f is increasing for every x* ∈ E. We show that if f: [0,1] → X is an increasing function with respect to a norming subset E of X* with uncountably many points of discontinuity and Q is a countable dense subset of [0,1], then (1) $\overline\{lin\{f([0,1])\}\}$ contains an order isomorphic copy of D(0,1), (2) $\overline\{lin\{f(Q)\}\}$ contains an isomorphic copy of C([0,1]), (3) $\overline\{lin\{f([0,1])\}\}/\overline\{lin\{f(Q)\}\}$ contains an isomorphic copy of c₀(Γ) for some uncountable set Γ, (4) if I is an isomorphic embedding of $\overline\{lin\{f([0,1])\}\}$ into a Banach space Z, then no separable complemented subspace of Z contains $I(\overline\{lin\{f(Q)\}\})$.},
author = {Artur Michalak},
journal = {Studia Mathematica},
keywords = {monotonic functions in Banach spaces},
language = {eng},
number = {2},
pages = {171-182},
title = {On monotonic functions from the unit interval into a Banach space with uncountable sets of points of discontinuity},
url = {http://eudml.org/doc/286607},
volume = {155},
year = {2003},
}

TY - JOUR
AU - Artur Michalak
TI - On monotonic functions from the unit interval into a Banach space with uncountable sets of points of discontinuity
JO - Studia Mathematica
PY - 2003
VL - 155
IS - 2
SP - 171
EP - 182
AB - We say that a function f from [0,1] to a Banach space X is increasing with respect to E ⊂ X* if x* ∘ f is increasing for every x* ∈ E. We show that if f: [0,1] → X is an increasing function with respect to a norming subset E of X* with uncountably many points of discontinuity and Q is a countable dense subset of [0,1], then (1) $\overline{lin{f([0,1])}}$ contains an order isomorphic copy of D(0,1), (2) $\overline{lin{f(Q)}}$ contains an isomorphic copy of C([0,1]), (3) $\overline{lin{f([0,1])}}/\overline{lin{f(Q)}}$ contains an isomorphic copy of c₀(Γ) for some uncountable set Γ, (4) if I is an isomorphic embedding of $\overline{lin{f([0,1])}}$ into a Banach space Z, then no separable complemented subspace of Z contains $I(\overline{lin{f(Q)}})$.
LA - eng
KW - monotonic functions in Banach spaces
UR - http://eudml.org/doc/286607
ER -