Let $X$ and $Y$ be normed spaces, $T \in B(X, Y)$ is bounded linear operator.
Def: $T$ is topologically injective iff it is homeomorphism onto image.
(a) Is it true that if $T$ is surjective, then $T^*$ is topologically injective? Is the converse true?
(b) Is it true that if $ImT$ is closed in $Y$, then $Im T^*$ is closed in $X^*$? Is the converse true?
For Banach spaces this is all true.
(c) Construct a condition on $T$ that is necessary and sufficient for $T^*$ to be topologically injective.
My attempts:
a) The first is not true.
$$ T: l_1 \rightarrow Im(l_1) \subset c_0$$ then $$ T^* : l_1 \rightarrow l_{\infty} $$ because $Im(l_1)$ with sup-norm is dense in $c_0$ but T* is not homeomorphism.
Converse is not true. Let $X$ - Banach space and $X_0$ is dense subspace and $T = id$. $$ T: X_0 \rightarrow X $$ $$ T^*: X^* \rightarrow X^* $$ and $T^*$ is topological isomorphism, but $T$ is not surjection.
b) Converse is not true again because we can choose dense subspace.
I think that there is no implication from the closed $ImT$ to closed $Im T^*$ (I don't know)
c) I know that is necessary $Im T$ is dense in $Y$. Then $T$ is injective, but $T^*$ must be open, then $T$ is isomorhism (bounded but not open (topological)). I think $T$ is isomorhism iff $T^*$ is topological injective.
Thank you so much!