I'm reading the book called "Optimization by Vector Space Methods" by David G. Luenberger. In the proof of the Lagrange Multiplier theorem, I don't understand the last part of the statement saying that "or, in an alternative notation,". I don't know why these two notations are equivalent.
I think that this just states that $$ A^* z_0^* = z_0^* A. $$ On the left-hand side, one is using the adjoint $A^* \colon Z^* \to X^*$ (if $A \colon X \to Z$), whereas $z_0^* A$ on the right-hand side should be interpreted as the concatenation of $z_0^* \colon Z \to \mathbb{R}$ with $A \colon X \to Z$. The identity just follows from the definition of the adjoint (or, depending on the precise definition, is just equal to the definition of the adjoint).
Further note that $\theta$ is the $0$ in the vector space where $f'(x_0)$ lives.