  In this paper, the module-algebra structures of $U_q(sl(m+1))$ on the quantum
$n$-space $A_q(n)$ are studied. We characterize all module-algebra structures
of $U_q(sl(m+1))$ on $A_q(2)$ and $A_q(3)$ when $m\geq 2$. The module-algebra
structures of $U_q(sl(m+1))$ on $A_q(n)$ are also considered for any $n\geq 4$.
