TITLE: Product of vector spaces
QUESTION [7 upvotes]: Let $V$ be a vector space over a fixed field $k$. Under what circumstances do we have $V\times V\cong V$? I think this should be true if $\mathrm{dim} \ V=\infty$, isn't it?

REPLY [3 votes]: I hope you know that $\dim_K V\times W=\dim_K V+\dim_K W$ for any two (finite dimensional) $K$-vector spaces $V,W$. 
Then it is clear that if $V$ is finite dimensional we can't have $\dim V\times V=\dim V$, unless $V=0$. 
But we can have $\dim V\times V=\dim V$ if $V$ is not finite dimensional: if $B\subset V$ is an infinite basis, then $|B|+|B|=|B|$, that is, there exists a bijection between the set $(B\times\{0\})\cup(\{0\}\times B)$ (which is in fact a basis in $V\times V$), and $B$ and this extends to an isomorphism of $K$-vector spaces between $V\times V$ and $V$.