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I need to prove the following proposition:

Any continuous linear functional on $H^1(\Omega)$ is of the form

$v\mapsto\displaystyle\int_\Omega\left\{\sum_{i=1}^nq_i\,\dfrac{\partial v}{\partial x_i}+q_0v\right\}$

with $q_i\in L^2(\Omega)$, $i=0,\dots n$.

Here $\Omega$ is an open bounded subset of $\mathbb{R}^n$ with a Lipschitz continuous boundary, and $H^1(\Omega)$ is the usual Sobolev space.

The book says that is a consequence from Banach Theorem, but I can't see the proof idea.

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    $\begingroup$ Try to see $H^1(\Omega)$ as a subset of $(L^2(\Omega))^{N+1}$ $\endgroup$ – Tomás Jan 4 '15 at 17:54
  • $\begingroup$ Sorry, I don't understand the hint. Can you help me with more details? $\endgroup$ – yemino Jan 4 '15 at 18:09
  • $\begingroup$ $H^1(\Omega)=\{v\in L^2(\Omega):\, \nabla u\in [L^2(\Omega)]^n\}$. I don't understant the hint because $H^1(\Omega)$ is an scalar space (it is not vector). $\endgroup$ – yemino Jan 4 '15 at 18:25
  • $\begingroup$ I tried to look the Hanh Banach theorem, but this theorem extends functional defined on a subspace to the whole space, so ? can't associate the Theorem with my problem $\endgroup$ – yemino Jan 4 '15 at 18:27
  • $\begingroup$ To give a function $u\in H^1(\Omega)$, is to give $N+1$ functions in $L^2(\Omega)$. $\endgroup$ – Tomás Jan 4 '15 at 18:37
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Ok, here it goes. We assume $F$ is an linear bounded operator over $H^1(\Omega)$

Let $E$ denote the space of $N+1$ fold $L^2(\Omega)$, i.e., $E(\Omega):=(L^2(\Omega))^{N+1}$. Then the operator $T$, from $H^1(\Omega)\to E(\Omega)$ is defined by $T[u]=(u,\partial_1 u,\partial_2u,\ldots,\partial_Nu)$ and we have $T[u]\in E(\Omega)$. Take $G:=T(H^1(\Omega))$ and $S:=T^{-1}$. Then, the linear operator $L$ over $G$ defined as $L(h):=\left<F,S(h)\right>$ is continuous because $F$ is continuous.

Now is where we use Hanh-Banach extension theorem. We have $G$ is a subspace over $E$ and hence we could extend $L$ from $G$ to whole $E$. Hence, $L$ is now a linear continuous operator over $E$. To conclude, we recall Riesz representation for $L^p$ and we obtain $v_0$, $v_1$, $v_2$...$v_n\in L^2(\Omega)$ such that $$ L(h)=\int_\Omega h_1v_0+\int_\Omega h_2v_1+\cdots+\int_\Omega h_{n+1}v_N\,dx $$ In particular, if $h\in G$ we have $$ L(h)=\left<F,S(h)\right>=\sum_{i=1}^N\int_\Omega \partial_i uv_i\,dx+\int_\Omega uv_0\,dx $$ as you expected.

Remark I don't think the boundary of $\Omega$ will matter, i.e., any open set would be good. Also, if you have $H_0^1$ you could set $v_0=0$.

Remark2 You can just extend $L$ by means of continuity but not H-B-E. And also notice that the sequence $(v_i)$ may not unique.

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