Basic question on dual spaces and adjoint in Hilbert spaces
$T^\dagger(f) = f \circ T$ gives $T^\dagger (Vx)=V(x)\circ T$ which is not the same as $V(T(x))$. (The left side is $ \langle Ty, x \rangle$ and the rigth side is $ \langle y, Tx \rangle$).
Basic question on dual spaces and adjoint in Hilbert spaces
$T^\dagger(f) = f \circ T$ gives $T^\dagger (Vx)=V(x)\circ T$ which is not the same as $V(T(x))$. (The left side is $ \langle Ty, x \rangle$ and the rigth side is $ \langle y, Tx \rangle$).