The Dual Space of an Asymmetric Normed Linear Space

Given an asymmetric normed linear space (X, q), we construct and study its dual space (X ∗ , q ∗). In particular, we show that (X ∗ , q ∗) is a biBanach semilinear space and prove that (X, q) can be identified as a subspace of its bidual by an isometric isomorphism. We also introduce and characterize the so-called weak* topology which is generated in a natural way by the relation between (X, q) and its dual, and an extension of the celebrated Alaoglu's theorem is obtained. Some parts of our theory are presented in the more general setting of the space LC (X, Y) of all linear continuous mappings from the asymmetric normed linear space X to the asymmetric normed linear space Y. In particular, we show that LC (X, Y) can be endowed with the structure of an asymmetric normed semilinear space and prove that it is a biBanach space if Y is so.