Suppose $V^{r}_{s}$ is $(r,s)$ type tensor space over number field $F$. We can form an algebra by sum all these tensor spaces, namely $$T(V)=\oplus_{r,s\geqslant 0}V_{s}^{r}=F\oplus V\oplus V^{\ast}\oplus \cdots$$
My confusion is: How can we direct sum all these different vector spaces? As I know, direct sum is an operation among subspaces of a vector space. Are these $V^{r}_{s}$ subspaces of some large vector space?
Direct sum $V_1 \oplus V_2 \oplus ... \oplus V_n$ of finitely many vector spaces $V_i$ is the same as their cartesian product $V_1 \times V_2 ... \times V_n$. They are different, though, when one considers direct sums and products of infinitely many vector spaces.