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

Artur Michalak

Artur Michalak

Studia Mathematica, Tome 157 (2003), p. 171-182 / Harvested from The Polish Digital Mathematics Library

Résumé

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) $\bar{l i n f ([0, 1])}$ contains an order isomorphic copy of D(0,1), (2) $\bar{l i n f (Q)}$ contains an isomorphic copy of C([0,1]), (3) $\bar{l i n f ([0, 1])} / \bar{l i n f (Q)}$ contains an isomorphic copy of c₀(Γ) for some uncountable set Γ, (4) if I is an isomorphic embedding of $\bar{l i n f ([0, 1])}$ into a Banach space Z, then no separable complemented subspace of Z contains $I (\bar{l i n f (Q)})$ .

Publié le : 2003-01-01

Zbl 1039.46012

EUDML-ID : urn:eudml:doc:286607

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Artur Michalak. On monotonic functions from the unit interval into a Banach space with uncountable sets of points of discontinuity. Studia Mathematica, Tome 157 (2003) pp. 171-182. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm155-2-6/