Limited type subsets of locally convex spaces

Abstract

Let $ 1\leq p\leq q\leq\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, \infty) $-limited sets, and if $ 1 < p < \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.

Keywords

• $ (p， q) $-limited set
• coarse $ p $-limited set
• $ p $-$ (v^\ast) $ set
• $ p $-convergent operator
• $ p $-barrelled space