Jesús A. Jaramillo Departamento de Análisis Facultad de Matemáticas Universidad Complutense de Madrid Spain E-mail address: jaramil@mat.ucm.es

Polynomial Topologies on a Banach Space

Abstract: Let X be a real Banach space. The weak polynomial topology w_{P} on X is defined as the weakest topology for which every polynomial on X is continuous. Then w_{P} is given by a family of translation-invariant semi-metrics, but in general it is not linear. Canonically related to w_{P}, we introduce the so called locally convex polynomial topology τ_{P} on X. We prove that τ_{P} is the finest locally convex topology which is coarser than w_{P}. Several properties of the seminorms defining τ_{P} are also studied. Finally, the convergence of sequences for w_{P} and τ_{P} is compared.