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(A)n with the additional assumption that A' is a manifold and !F is locally free was proved by Malgrange [12, p. 236, Problème 1]. In [9] Komatsu proved the following related result: if Sf is a coherent analytic sheaf on an n-dimensional complex manifold U such that Hn(U, Sf) = Q, then H\V, y)=0 for any open subset F of U (p. 83, Theorem 7). The author in [16] proved (A)n with the additional assumption that X is a manifold. The paper is divided into five sections. In §1 some Lemmas about Fréchet spaces and LF-spaces are proved. In §11 a duality concerning distributions with restricted supports is established. In §111 by partial normalizations and results of [13] the proof of the Main Theorem is reduced to the proof of (A)n with the additional assumption that X is reduced and normal and F is torsion-free. In §IV we prove by local resolutions of singularities that Hn(G, J^) = 0 for Gc ç x and also obtain a result on the approximation of (n— l)-cocycles with coefficients in &. §V sews up the proof of the Main Theorem. All complex spaces here are a-compact and are in the sense of Grauert [3, p. 9, Definition 2]. The structure sheaf of a complex space X is denoted by x<3 unless specified otherwise. The set of all singular points of X is denoted by o(X). The inverse image [4, p. 410, Definition 8] and the qth direct image [4, p. 413, Definition 9] of an analytic sheaf J5" under a holomorphic map / of complex spaces are denoted respectively byf'XF) and R"f(3P). If S? is a subsheaf of F, then R°f(&) is regarded as a subsheaf of R°f(F). A covering 11 of a complex space X is called a Stein covering if It is countable and every member of U is a Stein open subset. If Fis an open subset of X, then 111 Y={U e 111 £/<= Y} is called the restriction of 11

Analytic sheaf cohomology groups of dimension $n$ of $n-dimensional$ $complex$ $spaces.$