Linearity of an adjoint map

Question

In my class I was asked to solve the following task:

Suppose the map $\varphi : L \rightarrow M$ where L and M are linear spaces. Is the adjoint map $\varphi^* : M^* \rightarrow L^* $ linear?

This is my solution:

By definition of an adjoint map: $$(\varphi^*(f))(v) = f(\varphi(v))$$ for $f \in M^*$. Let's check for linearity:

By the definition of a funtional, we have $(f_1+f_2)(x)=f_1(x)+f_2(x)$ and $(\alpha f)(x)=\alpha f(x)$

1. $(\varphi^*(f_1+f_2))(v) =(f_1+f_2)(\varphi(v))=f_1(\varphi(v))+f_2(\varphi(v))=\varphi^*(f_1)+\varphi^*(f_2)$
2. $(\varphi^*(\alpha f))(v) = (\alpha f)(\varphi(v)) =\alpha f(\varphi(v))=\alpha \varphi^*(v)$

Hence, $\varphi^*$ is linear. The problem looked like pretty straight forward for me. But the true answer is "no, in general".

Is there a mistake in my solutoin?

Answer 1

Well, $\varphi$ is not linear, so its adjoint is not well defined, since $f\circ \varphi$ is not linear, and thus not an element of $L^*$.