• Oscar Valero

      Articles written in Proceedings – Mathematical Sciences

    • Quotient normed cones

      Oscar Valero

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      Given a normed cone (X, p) and a subconeY, we construct and study the quotient normed cone (X/Y,p) generated byY. In particular we characterize the bicompleteness of (X/Y, ‖·‖p,p) in terms of the bicompleteness of (X, p), and prove that the dual quotient cone ((X/Y)*, ¦¦ · ‖·‖p,p) can be identified as a distinguished subcone of the dual cone (X*, ¦¦ · ¦¦p, u). Furthermore, some parts of the theory are presented in the general setting of the spaceCL(X, Y) of all continuous linear mappings from a normed cone (X, p) to a normed cone (Y, q), extending several well-known results related to open continuous linear mappings between normed linear spaces.

    • Closed Graph and Open Mapping Theorems for Normed Cones

      Oscar Valero

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      A quasi-normed cone is a pair $(X, p)$ such that 𝑋 is a (not necessarily cancellative) cone and 𝑞 is a quasi-norm on 𝑋. The aim of this paper is to prove a closed graph and an open mapping type theorem for quasi-normed cones. This is done with the help of appropriate notions of completeness, continuity and openness that arise in a natural way from the setting of bitopological spaces.

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