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 $n$-weak amenability and strong double limit property Bull. Korean Math. Soc. 2005 Vol. 42, No. 2, 359-367 Published online June 1, 2005 A.R. Medghalchi and T. Yazdanpanah Teacher Training University, Persian Gulf University Abstract : Let ${\mathcal A}$ be a Banach algebra, we say that ${\mathcal A}$ has the strongly double limit property (SDLP) if for each bounded net $(a_{\alpha})$ in ${\mathcal A}$ and each bounded net $(a^*_{\beta})$ in ${\mathcal A}^*$, $\lim_{\alpha}\lim_{\beta} \left\langle a_{\alpha} , a^*_{\beta}\right\rangle = \lim_{\beta} \lim_{\alpha} \left\langle a_{\alpha} , a^*_{\beta} \right\rangle$ whenever both iterated limits exist. In this paper among other results we show that if ${\mathcal A}$ has the SDLP and ${\mathcal A}^{**}$ is $(n-2)$-weakly amenable, then ${\mathcal A}$ is $n$-weakly amenable. In particular, it is shown that if ${\mathcal A}^{**}$ is weakly amenable and ${\mathcal A}$ has the SDLP, then ${\mathcal A}$ is weakly amenable. Keywords : Banach algebra, weak amenability, Arens regular, $n$-weak amenability MSC numbers : 46H20, 46H40 Full-Text :