# $B$ be an Infinite dimensional Banach Space and $T:B\to B$ be an continuos operator

Let $B$ be an Infinite dimensional Banach Space and $T:B\to B$ be an continuous operator such that $T(B)=B$ and $T(x)=0\Rightarrow x=0$

which of the following is correct?

1. $T$ maps bounded sets into compact set

2. $T^{-1}$ maps bounded sets into compact set

3. $T^{-1}$ maps bounded sets into bounded set

4. $T$ maps compact sets into open set I have no idea how to do it.

-
Isn't this the bounded inverse theorem (en.wikipedia.org/wiki/Bounded_inverse_theorem), and hence the 3rd option? –  Jan Keersmaekers Sep 12 '13 at 12:48
Everything is always mapped into open sets. Should 4. say "onto open sets"? –  Jonas Meyer Sep 12 '13 at 17:53

$1.$ and $2.$ are not satisfied by $Tx=x$.
$3.$ $T^{-1}$ is bounded (as a consequence of the closed graph theorem).
$4.$ Take $K:=\{0\}$.
Davide could you tell me how $T^{-1}$ is bounded? –  miosaki Sep 13 '13 at 5:12