While reading Liviu Nicolaescu "Lectures on the Geometry of Manifolds" I came accross the notation, I'm not sure how to interpret. Hopefully, someone from this community will help to clarify things.
Let $E, F$ be two vector spaces over the field $R$. Consider the (infinite) direct sum $$T(E,F)=\bigoplus_{(e,f)\in E\times F}R.$$
I know what the direct sum of vector spaces is, but in this case I'm not sure how to interpret the infinite direct sum as is written above. I probably would be able to imagine the infinite sum $\bigoplus E\times F$. Could someone explain me what is the meaning of the above infinite sum?
In general, given a family $(V_i)_{i\in I}$ of vector spaces, the direct sum $\bigoplus_{i\in I} V_i$ consists of all families $(v_i)_{i\in I}$ with $v_i\in V_i$ for all $i\in I$, and such that $v_i=0$ for all but a finite number of $i$. In this case, $V_i=R$ for all $i$, and $I=E\times F$.