$V \cong V \oplus V$ as $K$ vector spaces I am not very sure about the triviality of this problem but I can't see the solution. Problem is If $V$ is a countable dimensional vector space over field $K$, then as $K$ vector spaces $V \cong V \oplus V$.

Suppose we have an indexing $v_1,v_2,\ldots$ of a basis of $V$. We may choose such an indexing by the fact that the vector space has countable dimension; by definition, dimension is the cardinality of the basis, so this means that a bijection of any basis with the natural numbers exists by assumption even in absence of the axiom of choice. Then an isomorphism is given by the linear map sending $v_{2k-1}\mapsto (v_k,0)$ and $v_{2k}\mapsto (0,v_k)$.