Let $X,Y$ be Banach spaces and $T:X\to Y$ a bounded linear application.
Let $T^*:Y^*\to X^*$ be its adjoint (i.e. $T^*g(x)=g(Tx)$, $\forall g\in Y^*, x\in X$ ).
If $||T^*g||\geq K||g||$ for some constant $K>0$, show that $T$ is surjective.
My work so far:
$\bullet$ from the lower bounded of $T^*$ we conclude that $T^*$ is a isomophism from $Y^*$ to $R(T^*)$, and so $R(T^*)$ is closed.
$\bullet$ As $T^*$ is injective, we also have that $R(T)$ is dense in $Y$.
But how can I show that $R(T)$ is closed?