My question is on the title , Let $\mathfrak{t}$ be densely defined form on $\mathcal{H}$. The operator $A$ associated with the form $\mathfrak{t}$ is defined by $Ax=u_x$ for $x\in D(A)$, where
$$ D(A)=\{x\in D(\mathfrak{t}) : \exists u_x\in \mathcal{H} \textrm{ such that } \mathfrak{t}[x,y]=\langle u_x,y \rangle \textrm{ for all } y\in D(\mathfrak{t}) \} $$
Now let,$\mathcal{H}=L^2(a,b)$, $\mathfrak{t}[f,g]=f(c)\overline{g(c)}$ and $D(\mathfrak{t})=C([a,b])$. I want to find the linear operator $A$ associated with this.
Maybe another way to write this question is like this: Find a linear operator $A$ s.t. $\langle Af,g \rangle = f(c)\overline{g(c)}$.
Only thing that comes to my mind is to say $A(f)(x)=\delta(x-c)f(c)$. But as we know $\delta(x)$ doesn't belong to $L^2$ space.
Thank you for your answers.