  In this note, we consider several notions related to the Grothendieck
property. Among them, we introduce the notion "unbounded Grothendieck property"
in a Banach lattice as an unbounded version of the known Grothedieck property
in the Banach space theory. Beside other results, surprisingly, we show that
spaces with the unbounded Grothendieck property are exactly the reflexive
Banach lattices.
