STRICT TOPOLOGY AS A MIXED TOPOLOGY ON LEBESGUE SPACES

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.