How can vector space $V\subsetneq W$ but $V\cong W$, where $V$ and $W$ don't have finite dimensions? Let $V$ and $W$ two vector spaces. If they have finite dimension, then $V\subset W$ and $V$ and $W$ have same dimension will imply that $V=W$. But I heard that in infinite dimensions, this doesn't hold anymore in the sense that you can have that $V$ is a proper subspace of $W$ (i.e. $V\subsetneq W$) and $V$ and $W$ are isomorphic. Could someone explain this fact please ? Because I really don't understand how is this possible.  

Vector spaces (for example over $\mathbb{R}$) are isomorphic if and only if they have the same dimension. The dimension is the cardinality of any basis.
Now let $V = R^\mathbb{N}$. Take any strict infinite subset $B'$ of the canonical basis $B$ of $V$ (for example, just remove one basis element). Then $|B'| = |B|$ so $V \cong \operatorname{span}(B')$
