Abstract
We study the local exactness of the \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\overline{\partial }$\end{document} operator in the Hilbert space l^2^ for a particular class of (0, 1)‐forms ω of the type \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\omega (z) = \sum _i z_i\omega ^i(z) d\overline{z}_i$\end{document}, z = (z~i~) in l^2^. We suppose that each function ω^i^ of class C^∞^ in the closure of the unit ball of l^2^, of the form \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\omega ^i(z) = \sum _k \omega ^i_k\left(z^k\right)$\end{document}, where \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbf N = \bigcup I_k$\end{document} is a partition of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbf N$\end{document} (card I~k~ < +∞) and z^k^ is the projection of z on \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbf C^{I_k}$\end{document}. We obtain a positive result (Theorem 1.1) when the sequence (card I~k~) is bounded. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim