# 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 *L*^{ 1 } (or more generally, produces the *p*-convexification of Banach lattices (of functions), [9], [99, pp. 53-54], [157]).

## Keywords

Banach Lattice Lattice Norm Continuous Linear Operator Order Ideal Concavity Property## Preview

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