STRICT TOPOLOGY AS A MIXED TOPOLOGY ON LEBESGUE SPACES

Abstract

AbstractLetXbe a locally compact space, and 𝔏∞0(X,ι) be the space of all essentially boundedι-measurable functionsfonXvanishing at infinity. We introduce and study a locally convex topologyβ1(X,ι) on the Lebesgue space 𝔏1(X,ι) such that the strong dual of (𝔏1(X,ι),β1(X,ι)) can be identified with$({\frak L}_0^\infty (X,\iota ),\|\cdot \|_\infty )$. Next, by showing thatβ1(X,ι) can be considered as a natural mixed topology, we deduce some of its basic properties. Finally, as an application, we prove thatL1(G) , the group algebra of a locally compact Hausdorff topological groupG, equipped with the convolution multiplication is a complete semitopological algebra under theβ1(G) topology.