Let $V$ be vector space of countable dimension over field $K$. Is it true that $V\cong V\oplus V$?

Question

I know that my question has been answered on StackExchange for the countably infinite case. I understand why this holds. However, I'm reading "A course in homological algebra" by Hilton,Stammbach and they assert in an exercise that $V\cong V\oplus V$ holds for countable dimensional (so possibly finite) vector spaces. I can't see why or even if this claim is true.

Please help if you have any idea if this is true or not.

Answer 1

For a finite-dimensional $V$ it is obviously wrong because $\dim(V \oplus V) = 2 \dim V > \dim V$.