Dimension of direct products of vector spaces

Question

Suppose $\{V_i\}_{i\in I}$ is a family of $k$-vector spaces. Is it possible to calculate $\dim\oplus_{i\in I} V_i$ and $\dim\prod_{i\in I}V_i$?

Answer 1

Assume $\dim V_i = \kappa_i$. Then the dimension of $\oplus_{i \in I} V_i$ is $\sum_{i \in I}\kappa_i$, where the sum denotes the sum of cardinals.

However, for the product, it seems like a difficult question, because there is no obvious basis, and I'm not certain. (If $I$ is finite, the product is of course isomorphic to the direct sum, so the same formula holds.)

The one comment I can make on this case is that if $V$ is a $k$-vector space with infinite dimension $\lambda$, then the cardinality of $V$ is $|V| = \sup(\lambda,|k|)$. So in cases where $\left|\prod V_i \right| > |k|$, the dimension of $\prod V_i$ must be equal to its cardinality. The cardinality of the set $\prod V_i$ can be calculated as $\prod |V_i| = \prod \sup(|k|,\kappa_i)$.