So I have a direct sum of vector spaces over the same field $V = \oplus_{i\in\mathbb{N}} V_i$. If I define a product for every two sets of vector spaces i.e. $\cdot:V_i\times V_j\rightarrow V_{i+j}$ how does this translate to the whole direct sum?
I suppose just $(\sum_{i}a_i)(\sum_{j}b_j) = \sum_{i,j}a_ib_j$
EDIT:
In particular, I am in the following case:
$T^rV^* := V^*\otimes\cdots\otimes V^*$ and $TV^* := \oplus_{r\in\mathbb{N}} T^rV^*$ then we define $T^rV^*\times T^sV^*\rightarrow T^{r+s}V^*:(\alpha,\beta)\mapsto\alpha\otimes\beta$