On some properties of separately increasing functions from \([0,1]^n\) into a Banach space. (English) Zbl 1312.46033
Summary: 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 \(x^*\in E\). A function \(f:[0,1]^m\to X\) is separately increasing if it is increasing in each variable separately. We show that if \(X\) is a Banach space that does not contain any isomorphic copy of \(c_0\) or such that \(X^*\) is separable, then for every separately increasing function \(f:[0,1]^m\to X\) with respect to any norming subset there exists a separately increasing function \(g:[0,1]^m\to \mathbb R\) such that the sets of points of discontinuity of \(f\) and \(g\) coincide.
MSC:
46B99 | Normed linear spaces and Banach spaces; Banach lattices |
28A78 | Hausdorff and packing measures |
26E20 | Calculus of functions taking values in infinite-dimensional spaces |