On monotonic functions from the unit interval into a Banach space with uncountable sets of points of discontinuity
Volume 155 / 2003
Abstract
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 $I(\overline{ \mathop{\rm lin}\{f(Q)\}})$.
