# What means norm $\|u'\|$ and $\|u\|$ in Sobolev space $W^{1,2}(I)$ and why is functional continuous?

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.

• These are the usual $L^2$ norms of $u$ and $u'$. – maxmilgram Mar 1 at 17:34
• The norm of this sobolev space ist just the sum of the $L^2$ norms of $u$ and $u'$. See here under "sobolev spaces with integer k". en.m.wikipedia.org/wiki/Sobolev_space – maxmilgram Mar 1 at 19:19
• Take $C=\max\{C_1, C_2 \}$ and you'll have your operator bounded – zorro47 Mar 1 at 21:49