I'm studying Dirk Werner's functional analysis book and I'm stuck with one direction of the following problem in chapter III.4:
$S \in L(Y',X')$ is an adjoint operator iff $S'(X) \subset Y$ (in Lemma III.4.3 German version of the book).
One direction is done in the book:
$\Rightarrow: (T''(i_X(x)))(y') = (i_X(x))(T'y')=T'y'(x)=y'(Tx)= (i_Y(Tx))(y')$.
For the other direction, I read something online (can't find the source right now), that says, if I can prove the identity $(i_X)' \circ i_{X'} = id_{X'}$ then I could use it to do the following: Let $S \in L(Y',X')$ be given and $S(X) \subset Y$. Then $S = S \circ id_{X'} = (i_X)' \circ i_{X'} \circ S = (i_X)' \circ S'' \circ i_{Y'} = (S' \circ i_X)' \circ i_{Y'} =^{(*)} (i_Y \circ T) \circ i_{Y'} = T' \circ (i_Y)' \circ i_{Y'} = T' \circ id_{Y'} = T'$.
$(*)$:because of the requirements exists such a T
But I don't manage to proof that $(i_X)' \circ i_{X'}= id_{X'}$.
I appreciate any help.
