Demi-linear duality

As is well known, there exist non-locally convex spaces with trivial dual and therefore the usual duality theory is invalid for this kind of spaces. In this article, for a topological vector space X , we study the family of continuous demi-linear functionals on X , which is called the demi-linear dual space of X . To be more precise, the spaces with non-trivial demi-linear dual (for which the usual dual may be trivial) are discussed and then many results on the usual duality theory are extended for the demi-linear duality. Especially, a version of Alaoglu-Bourbaki theorem for the demi-linear dual is established.