At first the author proves the following Theorem 1: Given \(1\leq r\), \(p_1,\dots,p_m1\) with \(\frac{1}{p_1}+ \frac{1}{p_2} +\cdots+ \frac{1}{p_m}=\frac{1}{r}\), let \((e_{jn})_n\) be a Schauder basis in \(\ell_{p_j}\) equivalent to the unit vector basis of this space \((j=1,\dots, m)\). If \(Y\) is the closed linear hull of \[ (e_{1n}\otimes e_{2n}\otimes\cdots\otimes e_{mn})_n \tag \(*\) \] in \(X:= \ell_{p_1} \widehat{\otimes}_\pi \ell_{p_2} \widehat{\otimes}_\pi\dots\widehat{\otimes}_\pi \ell_{p_m}\), then \(Y\) is isomorphic to \(\ell_r\), has a topological complement, and \((*)\) is a Schauder basis for \(Y\) equivalent to the unit vector basis of \(\ell_r\). From this follows that (1) \(T\) defined by \(Tg:= (g(e_{1n}, e_{2n},\dots, e_{mn}))_n\) is a continuous linear map from \({\mathcal L}(\ell_{p_1},\dots, \ell_{p_m})\) onto \(\ell_s\) and (2) \(S\) defined by \[ (Sg)(x_1,\dots, x_m):= \sum_{n=1}^\infty g(e_{1n},\dots, e_{mn}) \langle x_1,e_{1n}^*\rangle\langle x_2,e_{2n}^*\rangle\cdots \langle x_m,e_{mn}^*\rangle \] is a continuous projection in \({\mathcal L}(\ell_{p_1},\dots,\ell_{p_m})\) whose range is isomorphic to \(\ell_s\). Here \(1