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.  

A standard example is with the space of real polynomials $\mathbb{R}[x]$. If  you define a map $T:\mathbb{R}[x]\to \mathbb{R}[x]$ by $p(x)\to p(x^2)$ then it is an isomorphism between $\mathbb{R}[x]$ and the image of $T$. However $T(\mathbb{R}[x])$ is a proper subspace. 
