Here's a question in one of our exercises list :
Let $E$ be a normed vector space, $F$ be a Banach space and $T : E \to F$ a continuous linear application. Define $ E/\mathrm{Ker} (T) = \{ [x] \} $ where $[x]$ is the equivalence class of $x \in E$, i.e. $[x] = \{ x + y \, | \, y \in \mathrm{Ker} (T )\}$. This space is a normed vector space with the following norm : $ \Vert[x]\Vert = \inf \{ \Vert y \Vert \, | \, y \in [x] \}. $ Define $[T] : E / \mathrm{Ker}(T) \to \mathrm{Im}(T)$ by $[T]([x]) = T(x)$. Suppose that $\mathrm{Im}(T)$ is closed. Show that $[T]$ is an homeomorphism (i.e. its inverse is linear and continuous.
Now here's the deal ; this question turns out to be hard (and most probably false) because my teacher did a typo and hence forgot to mention that $E$ was supposed to be a Banach space for this question to work out. So I worked for a few days on it and only managed to show the following :
$[T]$ is an homeomorphism if and only if $E/\mathrm{Ker}(T)$ is a Banach space.
If $E/\mathrm{Ker}(T)$ is not Banach (i.e. not complete), there exists a Cauchy sequence $\{ [y_n] \}$ such that $\Vert [y_n] \Vert \to A > 0$ but $[T]([y_n]) \to 0$.
Now here is my question.
Can anyone find a counter example to show that this exercise is false (which is most probably the case), or prove it otherwise (which I have no hope in doing)?