This point has been giving me a lot of trouble. I am skipping around in learning functional analysis, and I've directly gone to the study of bounded operators without studying topology and basic Banach Space theory, in hopes to pick it up along the way. I'm doing okay with this....
Anyways, I am trying to show that given two Banach Spaces $X$ and $Y$, the operator $T$ in $L(X,Y)$ and its adjoint $T^{\ast}$ in $L(Y^{\ast},X^{\ast})$, then the map $T \mapsto T^\ast$ is an isometric isomorphism of $L(X,Y)$ into $L(Y^\ast,X^\ast)$. The proof is very straightforward, with one important detail, which is
$\Vert Tx\Vert_Y = \sup_{\Vert l \Vert \leq 1} |l(Tx)| \qquad \text{for } l \in Y^{\ast}$
the authors justify this as follows: "this equality uses a corollary of the Hahn-Banach Theorem".
I just don't see this! How does this follow?