Quasi-Banach Function Spaces

Abstract

Quasi-Banach spaces are an important class of metrizable topological vector spaces (often, not locally convex), [70], [83], [87], [88], [105], [135]; for quasi-Banach lattices we refer to [82, pp. 1116-1119] and the references therein. In the past 20 years or so, the subclass of quasi-Banach function spaces has become relevant to various areas of analysis and operator theory; see, for example, [29], [30], [32], [50], [59], [61], [87], [126], [152] and the references therein. Of particular importance is the notion of the p-th power X[_p], 0 ≺ p ≺ ∞, of a given quasi-Banach function space X. This associated family of quasi-Banach function spaces X[ _p ], which is intimately connected to the base space X, is produced via a procedure akin to that which produces the Lebesgue L _p -spaces from L¹ (or more generally, produces the p-convexification of Banach lattices (of functions), [9], [99, pp. 53-54], [157]).