V be a vector space over F. prove V is infinite dimensional if and only if there exists a proper subspace W of V which is isomorphic to V

Question

V be a vector space over F. prove V is infinite dimensional if and only if there exists a proper subspace W of V which is isomorphic to V

Answer 1

Let $B$ be a basis for $V$. If $V$ is infinite dimensional, i.e. $B$ is infinite, then there is a countably infinite subset $C\subseteq B$.

Define $f:C\to C$ by $f(c_n):=c_{n+1}$, extended to $f:B\to B$ by $f(b):=b$ for $b\notin C$, and extended linearly to $V$ by $ f(v)=f(\sum_i\alpha_ic_i+\beta_ib_i):=\sum_i\alpha_if(c_i)+\sum_i\beta_if(b_i)$. (All sums finite.)

Then $f(V)\subset V$ but $f(V)\cong V$. This follows from $f$ being linear and bijective.