The tensor algebra over a $n$-dimensional vector space $E$ is defined as the direct sum: $$\bigotimes E:=\bigoplus_{r,s>0}\Big(\bigotimes\nolimits_r^sE\Big)$$ An element of $\bigotimes E$ is written as the finite sum $$K=\sum_{r,s>0} K_r^s$$ with $K_r^s\in\bigotimes_r^sE$.
I only know how to sum $K+L$ with $K,L\in\bigotimes_r^sE$. How is defined $K_r^s$ and $K_{r'}^{s'}$ with $K_r^s\in\bigotimes_r^s E$ and $K_{r'}^{s'}\in\bigotimes_{r'}^{s'}E$?
How is valued $K$?
Many thanks!