STRICT TOPOLOGY AS A MIXED TOPOLOGY ON LEBESGUE SPACES

摘要

Let X be a locally compact space, and L(0)(infinity)(X, iota) be the space of all essentially bounded t-measurable functions f on X vanishing at infinity. We introduce and study a locally convex topology beta(1)(X, iota) on the Lebesgue space L(1)(X, iota) such that the strong dual of (L(1) (X, iota),beta(1)(X, iota)) can be identified with (L(0)(infinity)(X, iota), parallel to.parallel to(infinity)). Next, by showing that beta(1)(X, iota) can be considered as a natural mixed topology, we deduce some of its basic properties. Finally, as an application, we prove that L(1)(G), the group algebra of a locally compact Hausdorff topological group G, equipped with the convolution multiplication is a complete semitopological algebra under the beta(1) (G) topology.