The Strict Topology on the Discrete Lebesgue Spaces

Abstract

Let be a set and σ be a positive function on . We introduce and study a locally convex topology β 1 ( , σ ) on the space 1 ( , σ ) such that the strong dual of ( 1 ( , σ ), β 1 ( , σ )) can be identified with the Banach space (c 0 ( , 1/σ ), · ∞,σ ). We also show that, except for the case where is finite, there are infinitely many such locally convex topologies on 1 ( , σ ). Finally, we investigate some other properties of the locally convex space ( 1 ( , σ ), β 1 ( , σ )), and as an application, we answer partially a question raised by A. I. Singh ['L ∞ 0 (G) * as the second dual of the group algebra L 1 (G) with a locally convex topology', Michigan Math. J. 46 (1999), 143-150].

Strict Topology as a Mixed Topology on Lebesgue Spaces

“…By this observation, we prove some general properties of the locally convex space (L 1 (X, ι), β 1 (X, ι)). This generalizes and places our recent work [9] in a correct frame. In particular, we show that (L 1 (X, ι), β 1 (X, ι)) is bornological, barreled or metrizable space if and only if X is compact.…”

“…This topology was introduced in [13] for group algebras. For a recent study of this topology in the semigroup algebra setting see [10,11]; see also [9].…”

The space of vector‐valued integrable functions under certain locally convex topologies

“…The following theorem generalizes the main theorem of 20 with a totally different and much shorter proof; see also 11, 12.…”

“…The following key lemma shows that the topology β 1 (μ, E ) is, in fact, a type of strict topology. This is a corrected and generalized version of Proposition 4.9 in 11.…”

“…We denote this topology by β 1 (μ, E ). This topology in a special case was introduced in 20; see also 11, 12.…”