A topology on the Fremlin tensor product of locally convex-solid vector lattices

Abstract

Suppose that E and F are Banach lattices. It is known that there are several norms on the Fremlin tensor product \(E{\overline{\otimes }} F\) that turn it into a normed lattice; in particular, the projective norm \(|\pi |\) (known as the Fremlin projective norm) and the injective norm \(|\epsilon |\) (known as the Wittstock injective norm). Now, assume that E and F are locally convex-solid vector lattices. Although we have a suitable vector lattice structure for the tensor product E and F (known as the Fremlin tensor product and denoted by \(E{\overline{\otimes }}F\)), there is a lack of topological structure on \(E{\overline{\otimes }}F\), in general. In this note, we consider a linear topology on \(E{\overline{\otimes }}F\) that makes it into a locally convex-solid vector lattice, as well; this approach can be taken as a generalization of the projective norm of the Fremlin tensor product between Banach lattices.