Abstract
We claim, contrary to the linear case, that the lattice operations among harmonic functions are not necessarily monotone continuous in quasilinear harmonic spaces by showing the existence of a quasilinear harmonic space (X, H) in which there are harmonic functions un in H(X)(n=1, 2, …, ∞) with the following properties: the least harmonic majorant un∨0 and the greatest harmonic minorant un∧0 of un and 0 exist in H(X) for every n=1, 2, …, ∞; the sequence (un)1{≤}n<∞ is increasing and convergent to u∞ on X; the sequence (un∧0)1{≤}n<∞ converges increasingly to a harmonic function strictly less than u∞∧ 0 on X.
