I'm given the following operator on $L^2[0,1]$: $$Dom(T)=\{f\in L^2: f\text{ is continuous}\}$$ $$ Tf=f(0)$$ Where we consider $f(0)$ as a constant function. I wish to find its adjoint. I thus demand: $$\langle Tf,g\rangle=\int _0^1f(0)g=_{demand}\langle f,T^*g \rangle $$ By taking some continuous $f$ such that $f(0)=0$, we see that $ImT^*$ is orthogonal to the space of continuous functions such that $f(0)=0$. But since this subspace is dense in $L^2$, we conclude that $T^*=0$. The domain of $T$ is: $$Dom(T^*)=\{g\in L^2:f(0)\int_0^1g=\langle f,T^*g\rangle=0\}=sp\{1\}^{\perp}$$ Does this solution make sense? It seems strange to me that the adjoint is the $0$ map (although its domain is smaller), which makes me think that I misunderstand something important, so I'd appreciate any comment.
Thanks a lot in advance.