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$$f(0) = \int_0^1 (f'g_0' + fg_0) \text{ for all }f \in H^1(0, 1)?$$

Hint: 
$H^1(0,1)$ is a Hilbert space. If you can show that 
$$L : H^1(0,1) \to \mathbb R, \ \ \ L(f) = f(0)$$
is a bounded linear functional, the Riesz representation theorem tells you that 
$$L(f) = \langle f, g_0\rangle\ \ \ \forall f\in H^1(0,1),$$
which is exactly what you want. 
