Infinite direct sum not isomorphic to cartesian product? Let $\{V_i: i \in I\}$ be an infinite collection of non-trivial (i.e. not dimension $0$) vector spaces. I am trying to prove that $\bigoplus_{i \in I} V_i$ is not isomorphic to $\Pi_{i \in I} V_i$.
Here is what I tried: Towards a contradiction, assume there exists an isomorphism $f:\bigoplus_{i \in I} V_i \to \Pi_{i \in I} V_i$. Then choose a basis $B=\{b_j\}_j$ for $\bigoplus_{i \in I} V_i$. Observe that $C=\{f(b_j)\}_j$ must then be a basis for $\Pi_{i \in I} V_i$.
To reach the desired contradiction, I tried to come up with a diagonalization-style argument to choose a vector $w \in \Pi_{i \in I} V_i$ such that $w$ cannot possibly be a finite linear combination of $f(b_j)$ terms. But it's harder than I thought. For example, if we are lucky enough that each $f(b_j)$ is $0$ in all but finitely many coordinates, then we can just choose $w$ to have all of its coordinates nonzero and we're done. But in general $f(b_j)$ might not be so nice so it's harder.
(Please let me know if anything in my question needs clarification)
 A: In fact, the result you're trying to prove is not true in general! For example, taking $I=\mathbb{N}$ and setting all the $V_i$s to be the same for simplicity, we can have a vector space $V$ such that $$\prod_{i\in\mathbb{N}}V\cong \bigoplus_{i\in\mathbb{N}} V\cong V,$$ essentially for the same reason that we can have an infinite cardinal $\kappa$ such that $$\kappa^{\aleph_0}=\kappa.$$ This happens if e.g. $\dim(V)=2^{\aleph_0}$, so as a concrete example if our field of scalars is $\mathbb{R}$ we can take the vector space of all (not necessarily continuous!) functions $\mathbb{R}\rightarrow\mathbb{R}$.
(If the $V_i$s are all finite-dimensional vector spaces, then the result will hold since we will have $\dim(\bigoplus_{i\in I}V_i)=\vert I\vert$ but $\dim(\prod_{i\in I}V_i)=2^{\vert I\vert}.$)
Note that thinking in terms of dimension calculations simplifies things substantially; since a vector space is determined up to isomorphism (once we fix the field of scalars) by its dimension, which is a cardinal, for questions like this we're more-or-less just doing straight set-theoretic combinatorics in disguise.
