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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

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  • $\begingroup$ These are the usual $L^2$ norms of $u$ and $u'$. $\endgroup$ – maxmilgram Mar 1 at 17:34
  • $\begingroup$ That is not enough for my proffesor. Could you please write more about that norms or how they relate with boundedness? Thanks $\endgroup$ – Martin Květoň Mar 1 at 18:20
  • $\begingroup$ 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 $\endgroup$ – maxmilgram Mar 1 at 19:19
  • $\begingroup$ Take $C=\max\{C_1, C_2 \}$ and you'll have your operator bounded $\endgroup$ – zorro47 Mar 1 at 21:49

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