Pełczyński-type sets and Pełczyński’s geometrical properties of locally convex spaces

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 (sequential) Pełczyński’s property $V^\ast$ of order $(p,q)$, (sequential) Pełczyński’s property $V$ of order $(p,q)$, and Pełczyński’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.