Limited type subsets of locally convex spaces

Let $1\le p\le q\le \mathrm{\infty }.$ 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 $(1,\mathrm{\infty })$-limited sets, and if $1<p<\mathrm{\infty }$, then the class of coarse $p$-limited sets in $E$ coincides with the class of $p$- $(V^{\ast })$ sets of Pełczyński. We also generalize a known theorem of Grothendieck.