We have functional $F: W^{1,2}(I),I\langle 0,1\rangle \rightarrow \mathbb{R}$, I proved that functional is linear, I also counted that is bounded and I get this state:
$$|F(u)|\leq {C_1} \|u\|+{C_2}\|u'\|. $$
When I counted constats from integrals, I get: $C_1= \sqrt{3}$ and $C_2= \sqrt {2}$. But I dont know what means $\|u\|$, $\|u^\prime\|$these norms in space $W^{1,2}(I)$
When bounded then continuous. But this is not enough.
There should be something with Cauchy-Schwarz inequality. Please help. Thanks enter image description here
