Almost Summing Polynomials: A Coherent and Compatible Approach the from   the Infinite-Dimensional Holomorphy Viewpoint
  In this note we explore the notion of everywhere almost summing polynomials and define a natural norm which makes this class a Banach multi-ideal which is a 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 almost summing linear operators. Similar results are not valid for the original concept of almost summing polynomials, due to G. Botelho. 
Daniel Pellegrino and Joilson Ribeiro
math.FA