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Erschienen in:
2008 | OriginalPaper | Buchkapitel
Quasi-Banach Function Spaces
Erschienen in: Optimal Domain and Integral Extension of Operators
Verlag: Birkhäuser Basel
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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]).