Let $L,R: \Bbb R^{\infty} \rightarrow \Bbb R^\infty$ be the linear maps
$L(a_0, a_1,...) = (a_1, a_2,...)$ $R(a_0, a_1,...) = (0, a_0,...)$
Prove that the set of linear maps $R^kL^l$ is linearly independent.
I have struggled with the problem for a while. I assumed that $c_1R^{k1}L^{l1}+\dots+c_nR^{kn}L^{ln} = 0$ and if $c_1+\dots+c_n \neq 0$, I can conclude that it is contradicted, but I haven't figured out if $c_1 + \dots +c_n =0$.
You can definitely adopt a different method from mine. Any idea would be appreciated!