I know that function evaluation in $H^1(0,1)$ is continuous (see, e.g., Is the Delta distribution a continuous functional on $H^1(\mathbb R)$).
So, $\delta_x : H^1(0,1)\to\mathbb C$ is a continuous operator. Since it has finite-dimensional range, it is compact. In other words, $\delta_x$ is $D$-compact in $L^2(0,1)$, where $Df = f'$, $f\in H^1(0,1)$.
As $D$-compact operators have $D$-bound zero, for any $\varepsilon>0$, I should find an $a > 0$ such that $$ |f(x)|\,\le\,a\|f\|_2 + \varepsilon\|f'\|_2,\quad f\in H^1(0,1). $$ But I can't find such $a$ for given $\varepsilon$. Can anyone help me out?