Limited type subsets of locally convex spaces

Let 1 ≤ p ≤ q ≤ ∞. Being motivated by the classical notions of limited, p-limited, and coarse p-limited subsets of a Banach space, we introduce and study (p, q)-limited subsets and their equicontinuous versions and coarse p-limited subsets of an arbitrary locally convex space E. Operator characterizations of these classes are given. We compare these classes with the classes of bounded, (pre)compact, weakly (pre)compact, and relatively weakly sequentially (pre)compact sets. If E is a Banach space, we show that the class of coarse 1-limited subsets of E coincides with the class of (1, ∞)-limited sets, and if 1 < p < ∞, then the class of coarse p-limited sets in E coincides with the class of p-(V^∗) sets of Pełczyński. We also generalize a known theorem of Grothendieck.