Q:

Isomorphisms between infinite dimentional spaces

Let $V$ be an infinite dimensional vector space. Can we find an isomorphism between $V$ and $V \oplus V$. If the answer is positive then how this isomorphism can be constructed?

A:

Yes (given the Axiom of Choice). Let $\{v_i\}_{i\in I}$ be a basis of $V$. As $I$ has infinite cardinality, $|I|=|I|+|I|$, i.e. there exists a bijection $f\colon (I\times\{0\})\cup (I\times\{1\})\to I$.
Then $(v_i,0)\mapsto v_{f(i,0)}$, $(0,v_i)\mapsto v_{f(i,1)}$ is a bijection between a basis of $V\oplus V$ and the given basis of $V$, which extends linearly to an isomorphism of vector spaces.
Note however that per reference ot the Axiom of Choice this isomorphism cannot in full sense be called "constructed".

