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

Artur Michalak¹

¹ Faculty of Mathematics and Computer Science Adam Mickiewicz University Umultowska 87 61-614 Poznań, Poland

Studia Mathematica, Tome 155 (2003) no. 2, pp. 171-182

Résumé

We say that a function $f$ from $[0,1]$ to a Banach space $X$ is increasing with respect to $E\subset X^*$ if $x^*\circ f$ is increasing for every $x^*\in E$. We show that if $f:[0,1]\to X$ 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) $\overline{\mathop{\rm lin}\{f([0,1])\}}$ contains an order isomorphic copy of $D(0,1)$, (2) $\overline{\mathop{\rm lin}\{f(Q)\}}$ contains an isomorphic copy of $C([0,1])$, (3) $\overline{ \mathop{\rm lin}\{f([0,1])\}}/\overline{ \mathop{\rm lin}\{f(Q)\}}$ contains an isomorphic copy of $c_0({\mit\Gamma})$ for some uncountable set ${\mit\Gamma}$, (4) if $I$ is an isomorphic embedding of $\overline{\mathop{\rm lin}\{f([0,1])\}}$ into a Banach space $Z$, then no separable complemented subspace of $Z$ contains $I(\overline{ \mathop{\rm lin}\{f(Q)\}})$.

Détail

DOI : 10.4064/sm155-2-6

Keywords: say function banach space increasing respect subset * * circ increasing every * increasing function respect norming subset * uncountably many points discontinuity countable dense subset nbsp overline mathop lin contains order isomorphic copy nbsp overline mathop lin contains isomorphic copy nbsp overline mathop lin overline mathop lin contains isomorphic copy mit gamma uncountable set mit gamma nbsp isomorphic embedding overline mathop lin banach space separable complemented subspace contains overline mathop lin