Q:

Linear bijection between a normed vector space and its proper subspace

Let $X,Y$ be normed vector spaces such that $X$ is a proper subspace of $Y$ (i.e., $X$ is a subspace of $Y$ and $X\neq Y$), and $X$ is dense in $Y$. Is there a linear bijection $T$ from $X$ to $Y$? If it does exist, can $T$ and $T^{-1}$ both be bounded?

I only know that in finite dimensional spaces, this is impossible. But I have no idea how to come up with an example with such $T$ even in indefinite dimensional spaces (or disprove the existence).

A:

First question. In general the desired bijection does not exist. Consider dense subspace $X$ of countable dimension and compete separable Banach space $Y$ (for example $X=c_{00}$, $Y=\ell_2$), then cardinality of Hamel basis of $X$ is $\aleph_0$, of $Y$ is $\mathfrak{c}$. Since cardianality of Hamel basis is preserved under linear bijections, then there is no bijective linear operator between $X$ and $Y$.
Second question. I don't know the answer but the following can be stated. If such $T$ exists then $Y$ is incomplete. Indeed $T$ is an isomorphism, so if $Y$ is complete then $X$ is complete too. Then you can easily show that $X=Y$. So counterexamples (if they exist) are among incomplete normed spaces.

