This morning I was thinking at the following (simple) fact. Let us consider $[0, 1] \to \mathbb{R}$ functions and define a linear functional
$F(u)=u(1)-u(0).$
$F$ is not continuous on $L^2(0, 1)$ (in fact, it is not even defined everywhere), but it is continuous on $H^1(0, 1)$:
\lvert F(u) \rvert \le \int_0^1\lvert u'(x)\rvert\, dx\le \lVert u \rVert_{H^1}.
How would you give an intuitive explanation of this phenomenon? In what sense is Sobolev norm more restricting, so that a "bad" $L^2$ functional turns out to be a "good" one on $H^1$?