Abstract. The Bohnenblust-Hille inequality was obtained in 1931 and (in the case of real scalars) asserts that for every positive integer N and every m-linear mapping T :T (e i 1 , ..., e im )for some positive constant Cm. Since then, several authors obtained upper estimates for the values of Cm. However, the novelty presented in this short note is that we provide lower (and non-trivial) bounds for Cm.

Let X be a Banach space. We prove that, for a large class of Banach or quasi-Banach spaces E of X-valued sequences, the sets E − q∈Γ ℓ q (X), where Γ is any subset of (0, ∞], and E − c 0 (X) contain closed infinite-dimensional subspaces of E (if non-empty, of course). This result is applied in several particular cases and it is also shown that the same technique can be used to improve a result on the existence of spaces formed by norm-attaining linear operators.

This paper has a twofold purpose: to prove a much more general Dvoretzky-Rogers type theorem for absolutely summing polynomials and to introduce a more convenient norm on the space of everywhere summing polynomials.

In this note we solve, except for extremely pathological cases, a question posed by Puglisi and Seoane-Sepúlveda on the lineability of the set of bounded linear non-absolutely summing operators. We also show how the idea of the proof can be adapted to several related situations.

We provide (for both the real and complex settings) a family of constants, (Cm) m∈N , enjoying the Bohnenblust-Hille inequality and such that lim m→∞ Cm C m−1 = 1, i.e., their asymptotic growth is the best possible. As a consequence, we also show that the optimal constants, (Km) m∈N , in the Bohnenblust-Hille inequality have the best possible asymptotic behavior. Besides its intrinsic mathematical interest and potential applications to different areas, the importance of this result also lies in the fact that all previous estimates and related results for the last 80 years (such as, for instance, the multilinear version of the famous Grothendieck Theorem for absolutely summing operators) always present constants Cm's growing at an exponential rate of certain power of m.2010 Mathematics Subject Classification. 46G25, 47L22, 47H60.