Exist unique $g_0 \in H^1(0, 1)$ such that $f(0) = \int_0^1 (f'g_0' + fg_0) \text{ for all }f \in H^1(0, 1)$? The mapping $f \mapsto f(0)$ from $H^1(0, 1)$ into $\mathbb{R}$ is a continuous linear functional on $H^1(0, 1)$. Does there exist a unique $g_0 \in H^1(0, 1)$ such that$$f(0) = \int_0^1 (f'g_0' + fg_0) \text{ for all }f \in H^1(0, 1)?$$

Existence is demonstrated using $g(x)=-\cos(x-1)/\sin(1)$, which is the unique solution of
$$
        -g''+g = 0 \\
         g'(0)=-1,\;g'(1)=0.
$$
Then
$$
   \int_{0}^{1}\{f'(x)g'(x)+f(x)g(x)\}dx \\
   = \left.f(x)g'(x)\right|_{0}^{1}+\int_{0}^{1}f(x)\{-g''(x)+g(x)\}dx =f(0).
$$
Uniquess follows from the Riesz Representation Theorem.
