We investigate the summability of the coefficients of $m$-homogeneous polynomials and $m$-linear mappings defined on $\ell_{ p}$-spaces. In our research we obtain results on the summability of the coefficients of $m$-linear mappings defined on $\ell_{ p_{ 1}} \times \cdots \times \ell_{ p_{ m}}$. The first results in this respect go back to Littlewood and Bohnenblust and Hille (for bilinear and $m$-linear forms on $c_{ 0}$) and Hardy and Littlewood and Praciano-Pereira (for bilinear and $m$-linear forms on arbitrary $\ell_{ p}$-spaces). Our results recover and in some case complete these old results through a general approach on vector valued $m$-linear mappings.
Categories: math.FA