  For $1\leq p\leq q\leq\infty$ and a locally convex space $E$, we introduce
and study the $(V^\ast)$ subsets of order $(p,q)$ of $E$ and the $(V)$ subsets
of order $(p,q)$ of the topological dual $E'$ of $E$. Using these sets we
define and study the (sequential) Pe{\l}czy\'{n}ski's property $V^\ast$ of
order $(p,q)$, the (sequential) Pe{\l}czy\'{n}ski's property $V$ of order
$(p,q)$, and the Pe{\l}czy\'{n}ski's property $(u)$ of order $p$ in the class
of all locally convex spaces. To this end, we also introduce and study several
new completeness type properties, weak barrelledness conditions, Schur type
properties, the Gantmacher property for locally convex spaces, and
$(q,p)$-summing operators between locally convex spaces. Applications to some
classical function spaces are given.
