My notes give the cartesian product of the sets $X_1, . . . , X_n$ as $$X_1 × · · · × X_n = \{(x_1, . . . , x_n) : x_i ∈ X_i for 1 \le i \le n\}$$ I believe we can think of a vector space $V$ where $V=F^n$ as the cartesian product of $n$ copies of a set $F$, where $F$ is the ground field.
Also, given a field $F$ and F-vector spaces $V_1, . . . , V_n$, it can be shown that the cartesian product $V_1 × · · · × V_1$ is also an F-vector space if we define addition and multiplication by scalars componentwise. I would understand this to look like: $$V_1 × · · · × V_n = \{(v_1, . . . , v_n) : v_i ∈ V_i for 1 \le i \le n\}$$ My notes then give a different notation of the above cartesian product as: $$V_1 ⊕ · · · ⊕ V_n$$ and we refer to it as the product or the direct sum or the external direct sum. Is this simply $$V_1 ⊕ · · · ⊕ V_n = \{(v_1+ . . . + v_n) : v_i ∈ V_i for 1 \le i \le n\}$$? If so, how are these equivalent sets? Thanks!