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Yayın Some forms of the banach-steinhaus theorem In the locally convex cones(Maltepe Üniversitesi, 2009) Ranjbari, AsgharA cone is a set P endowed with an addition and a scalar multiplication for non-negative real numbers. The addition is associative and commutative, and there is a neutral element 0 ? P. For the scalar multiplication the usual associative and distributive properties hold. We have 1a = a and 0a = 0 for all a ? P. A preordered cone is a cone with a reflexive transitive relation ? which is compatible with the algebraic operations. A subset V of the preordered cone P is called an (abstract) 0-neighborhood system, if V is a subcone without zero directed towards 0. We call (P, V) a full locally convex cone, and each subcone of P, not necessarily containing V, is called a locally convex cone. We require the elements of a locally convex cone to be bounded below, i.e. for every a ? P and v ? V we have 0 ? a + ?v for some ? > 0. We verify some forms of the Banach-Steinhaus Theorem in the locally convex cones.
