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

Question

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.

Answer 1

$1.$ and $2.$ are not satisfied by $Tx=x$.

$3.$ $T^{-1}$ is bounded (as a consequence of the closed graph theorem).

$4.$ Take $K:=\{0\}$.