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

For 1≤p≤q≤∞ and a locally convex space E, we introduce and study the (V∗) 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∗ 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.