Authors:Vakhtang Kokilashvili and Alexander Meskhi
García-Cuerva , J. and Rubio de Francia , J.L., Weighted norm inequalities and related topics , North-Holland Mathematics Studies, Vol. 116, Mathematical Notes, 104, (North-Holland Publishing Co., Amsterdam, 1985
If (X, p) and (Y, q) are two asymmetric normed spaces, the set LC(X, Y) of all continuous linear mappings from (X, p) to (Y, q) is not necessarily a linear space, it is a cone. If X and Y are two Banach lattices and p and q are, respectively, their associated asymmetric norms (p(x) = ‖+‖, q(y) = ‖y+‖), we prove that the positive operators from X to Y are elements of the cone LC(X, Y). We also study the dual space of an asymmetric normed space and finally we give open mapping and closed graph type theorems
in the framework of asymmetric normed spaces. The classical results for normed spaces follow as particular cases.