For a vector space $V$ of infinite dimension, to show that $V\oplus V$ is isomorphic to $V$ is to show that there exists an invertible linear transformation between $V \oplus V $ and $V$.

Every vector space have a basis.
If $B$ is the infinite set of basis for $V$, then the set $B\oplus0 \cup 0\oplus B$ is the basis for $V \oplus V$. Using axiom of choice the cardinality $|B\oplus0 \cup 0\oplus B|= |B\oplus0|+|0 \oplus B|=|B|+ |B| = \max\{|B|,|B|\}= |B|$ and then I was trying to argue that this implies that there is an isomorphism between the bases of $V\oplus V$ and $V$, so they are isomorphic. But I'm not sure how $|B|+|B|=\max\{|B|,|B|\}= |B|$ is true by the axiom of choice. How could we show that if $V$ is infinite dimensional vector space, then $V\oplus V \cong V$?