Asymmetric locally convex spaces (Q2368377)

An asymmetric seminorm \(p\) on a real vector space \(X\) is a positive sublinear functional; i.e., \(p\) is subadditive, but satisfies \(p(tx)=tp(x)\) for \(x \in X\) only when \(t \geq 0\). Spaces with asymmetric norms have been studied since about 2000; they have applications in the complexity theory of algorithms, see e.g. \textit{L. M.\ García-Raffi}, \textit{S.\ Romaguera} and \textit{E. A.\ Sánchez Pérez} [Math.\ Comput.\ Modelling 36, 1--11 (2002; Zbl 1063.68057)]. In the article under review, the theory of asymmetric locally convex spaces is developed parallel to the theory of the `usual' locally convex spaces. There are some differences: The topology generated by an asymmetric norm is not always Hausdorff, and the dual of an asymmetric locally convex space need not be a linear subspace of the algebraic dual, but is merely a convex cone. The Hahn--Banach theorem remains a very important tool, and there is an analog of the Alaoglu--Bourbaki theorem. The author also proves asymmetric analogs of the classical separation theorems of Eidelheit and Tukey and of the Krein--Milman theorem. In the last section of the article, the asymmetric weak topology is investigated.