We invent the new notion of coordinatewise multiple summing operators in Banach spaces, and use it to study various vector valued extensions of the well-know Bohnenblust-Hille inequality (which originally extended Littlewood's 4/3-inequality). Our results have application on the summability of monomial coefficients of m-homogeneous polynomials P : ∞ → p , as well as for the convergence theory of products of vector valued Dirichlet series.

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Abstract. We estimate the 1-norm N n=1 an of finite Dirichlet polynomials N n=1 ann −s , s ∈ C with coefficients an in a Banach space. Our estimates quantify several recent results on Bohr's strips of uniform but non absolute convergence of Dirichlet series in Banach spaces.