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Continuous operators on asymmetric normed spaces

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Abstract

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.

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Authors and Affiliations

  1. Departamento Matemática Aplicada, IUMPA, Universidad Politécnica de Valencia, Camino de Vera s/n, 46022, Valencia, Spain

    C. Alegre

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  1. C. Alegre
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Correspondence to C. Alegre.

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The author acknowledges the support of the Ministerio de Educación y Ciencia of Spain and FEDER, under grant MTM2006-14925-C02-01 and Generalitat Valenciana under grant GV/2007/198.

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Alegre, C. Continuous operators on asymmetric normed spaces. Acta Math Hung 122, 357–372 (2009). https://doi.org/10.1007/s10474-008-8039-0

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