Injective and surjective maps between infinite dimensional vector spaces Suppose we have two infinite dimensional, topological vector spaces $V$ and $W$.
$\underline{\text{Question 1}:}$ Suppose $\phi_1\colon V\to W$ and $\phi_2\colon W\to V$ are injective continuous linear maps. Are $V$ and $W$ isomorphic?
$\underline{\text{Question 2}:}$ Suppose $\psi_1\colon V\to W$ and $\psi_2\colon W\to V$ are surjective continuous linear maps. Are $V$ and $W$ isomorphic?
If they are not isomorphic, how are they related?
My attempt to question number $1$ is as follows :
Consider $V=\ell_1$ and $W=\ell_2$ and
$$\phi_1\colon\ell_1\hookrightarrow\ell_2$$
$$\text{and}$$
$$\begin{align}\phi_2\colon\ell_2&\to\ell_1\\(x_1,x_2,x_3,\ldots)&\mapsto\left(\frac{x_1}{1},\frac{x_2}{2},\frac{x_3}{3},\ldots\right)\end{align}$$
are injections and continuous and $\ell_1$ and $\ell_2$ are not isomorphic.
Is my argument okay? What can we say about the second question? What if we omit the continuity of each of the linear maps?
 A: The answer to question 1 is fine.
For question 2, the spaces do not have to be isomorphic. There might be a more elemntary example, but this is what came to mind.
Let $V=\ell^\infty(\mathbb N)$, and $W=\ell^\infty(\mathbb N)/c_0$. Let $\psi_1:V\to W$ be the quotient map, so continuous and onto. To define $\psi_2:W\to V$ we will take advantage of the fact that $V$ and $W$ are also algebras (C$^*$-algebras, to be precise); consider a countable family $\{p_n\}$ of pairwise orthogonal projections in $V$ such that $p_n-p_m\not\in c_0$ if $n\ne m$ (construction below, at the end).
For each $n$, let $\beta_n\in W^*$ be a bounded linear functional with $\|\beta_n\|=1$ and $\beta_n(p_k)=\delta_k$ (these are easy to obtain because we can represent $W$ as operators on a Hilbert space and as the $p_n$ are pairwise orthogonal projections there exist orthonormal $\{\xi_n\}$ with $p_n\xi_n=\xi_n$; then put $\beta_n(x)=\langle x\xi_n,\xi_n\rangle$). Then we define
$$
\psi_2(a+c_0)=\big(\beta_n(a+c_0))_n.
$$
This map is clearly linear and contractive. Given $r\in\ell^\infty(\mathbb N)$ we can take $a+c_0=\sum_nr_np_n+c_0$ and then $\psi_2(a+c_0)=r$. So $\psi_2$ is surjective.
Finally, it is known that $W$ does not embed in $V$, so in particular they cannot be isomorphic.

Construction of the $p_n$.
Let $\{s_n\}$ be an enumeration of the prime numbers. Define
$$
p_n=\sum_ke_{s_n^k}.
$$
Since all power of primes are distinct, no entry is $1$ in more than one $p_n$. This gives us $p_n-p_m\not\in c_0$ if $n\ne m$.
