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?
Best Answer
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 $n\ne m$.