A Note on Scalar-valued Nonlinear Absolutely Summing Mappings

w,q = supφ∈BX́ ( ∑k j=1 | φ(xj) | ) 1 q . This is a natural generalization of the concept of (p; q)-summing operators and in the last years has been studied by several authors. The infimum of the L > 0 for which the inequality holds defines a norm ‖.‖as(p;q) for the case p ≥ 1 or a p-norm for the case p < 1 on the space of (p; q)-summing homogeneous polynomials. The space of all m-homogeneous (p; q)-summing polynomials from X into Y is denoted by Pas(p;q)( X ;Y ) (Pas(p;q)( X) if Y = K). When p = q m we have an important particular case, since in this situation there is an analogous of the GrothendieckPietsch Domination Theorem. The ( q m ; q)-summing m-homogeneous polynomials fromX into Y are said to be q-dominated and this space is denoted by Pd,q( X ;Y ) (Pd,q( X) if Y = K). To denote the Banach space of all continuous m-homogeneous polynomials P from X into Y with the sup norm we use P(X,Y ) (P(X), if Y is the scalar field). Analogously, the space of all continuous m-linear mappings from X1 × ... × Xm into Y (with the sup norm) if represented by L(X1, ..., Xm;Y ) (L(X1, ..., Xm) if Y = K). The concept of absolutely summing multilinear mapping follows the same pattern (for details we refer to [5]). Henceforth every polynomial and multilinear mapping is supposed to be continuous and every Lp-space is assumed to be infinite-dimensional. A natural problem is to find situations in which the space of absolutely summing polynomials coincides with the space of continuous polynomials (coincidence situations). When Y is the scalar-field, these situations are not rare as we can see on the next two well known results: