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In this note we explore the notion of everywhere almost summing polynomialsand define a natural norm which makes this class a Banach multi-ideal which isa holomorphy type (in the sense of L.Nachbin) and also coherent and compatible(in the sense of D. Carando, V. Dimant and S. Muro) with the notion of almostsumming linear operators. Similar results are not valid for the originalconcept of almost summing polynomials, due to G. Botelho.

Almost Summing Polynomials: A Coherent and Compatible Approach the from the Infinite-Dimensional Holomorphy Viewpoint