Functions determining locally solid topological Riesz spaces continuously embedded in $L^0$

Abstract

We present a class of non-negative functions, acting on a solid vector subspace X of L 0 , enjoying the following property: each member of the class determines on X a locally solid topological Riesz space structure which is continuously embedded into L 0 . These functions are neither necessarily monotone, nor subadditive. Special instances are provided by function norms and quasi-norms on X .

LpSpaces in vector lattices and applications

“…In this paper some fundamental properties of L p spaces in the vector lattice setting are investigated, continuing a research initiated by the authors in [12-14, 16, 21, 22] and developed later in [15]. The range of the involved functions is a vector lattice endowed with filter/ideal convergence (for a related literature, see also [3,[17][18][19][20]25,27]). Thanks to the triangle inequality, it is possible to view the space L p as a metric space endowed with a distance of the type d(f, g) = f −g p .…”