For Banach space $l_p(l_q)$, where $p,q\geq 1$, does the following vectors form a Schauder basis of it, say a rearrangement of all $v_{m,n}=(0,...,0,e_m,0,...)$, where $e_m$ occur in $n$th coordinate and $\{e_m\}$ is the classical unit vector basis of $l_q$?
If yes, is the basis symmetric? If not, Is there a basis for this Banach space. In addition, If $X$ has a Schauder basis, does necessarily $l_p(X)$ has a basis?
Thank you for all helps!