  Unbounded convergences have been applied successfully to locally solid
topologies on vector lattices. In the present paper, we first expose several
properties of various classes of Riesz pseudonorms on vector lattices. We
accomplish this by abstracting some generalities of the norm of an AM-space
with strong norm unit to locally solid topologies induced by a pseudonorm. By
using these classes of pseudonorms, we study compactness properties of
operators (not necessarily linear) between locally solid (not necessarily
Hausdorff) topologies. We study new classes of operators such as pseudonorm
compact, pseudo-semicompact and pseudo-AM-compact operators as well as the
classical Montel operators.
