Let $X$ and $Y$ be Banach spaces, $X^*$,$Y^*$ be their dual spaces. $L(X,Y)$ denote the space of continuous linear functions from $X \to Y$. For each $T:X\to Y$, define $$T':Y^*\to X^*$$ such that $$l(Tx)=(T'l)(x)$$ For all $l \in Y^*$ and $x \in X$
Then we can show that $T'$ is a continuous linear function in $L(Y^*,X^*)$, and call it the Banach dual of $T$. Also, the map $T \to T'$ is an isometry.
However, in Michael Reed's functional analysis page $187$, it said that the map $T \to T'$ is an isometric isomorphism. I can not see why the map $T \to T'$ is surjective.