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My question is about functionals on $W_{1,p}(\Omega)$ spaces, $\Omega$ is contained in $\mathbb R^n$

I am trying to figure out if there is a way to characterize all linear functionals on the above space.

Is there any version of Riesz representation theorem in general Banach space?

Guy Fsone
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Lin
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  • Although, you cannot achieve that in $W_{1,p}$, there are Riesz-like results for $\mathcal{L}p$ and $\ell_p$ spaces. Let $X=\mathcal{L}_p(\Omega, \mathscr{F},\mathrm{P})$. According to D.S. Bridges, Foundations of real and abstract analysis, Springer, 1998, p. 202, we may identify all linear continuous functionals $f:X\to \mathbb{R}$ by elements $\tilde{f}\in X^*$ such that $f(x) = \int\Omega \tilde{f}(\omega)x(\omega)\mathrm{P}(\mathrm{d}\omega)$. – Pantelis Sopasakis Dec 11 '16 at 13:54

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There is no Riesz representation theorem that would say something about a general Banach space $X$. For such a space, it is customary to denote the space of all continuous linear functionals as $X^*$, which is called the dual space of $X$. One can try to give a concrete description of the elements of $X^*$, but the success depends on what $X$ is. For many function spaces the structure of the dual is already known, as searching for "dual of ... space" will show you.

Specifically, the dual space of the Sobolev space $W^{1,p}$ is described by Theorem 3.9 of the book Sobolev spaces by Adams.

For every $L\in (W^{1,p}(\Omega))^*$ there exist elements $v_0,v_1,\dots,v_n\in L^{p'}(\Omega)$ (where $n$ is the dimension and $p'=p/(p-1)$) such that $$L(u) = \int_\Omega \left(uv_0+\sum_{k=1}^n \frac{\partial u}{\partial x_k} v_k\right)$$

The structure of the dual is nicer if one restricts the attention to $W_0^{1,p}(\Omega)$, because integration by parts does not produce boundary terms then. The dual of $W_0^{1,p}(\Omega)$ is naturally identified with $W^{-1,p'}(\Omega)$, a Sobolev space of negative order of smoothness.

See also: Dual space of the sobolev spaces.

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  • Do you know any space of functions $X$ such that $$X^\star=W^{1,1}(\Omega)?$$ – Tomás Mar 03 '15 at 15:26
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    There is no such $X$. A separable dual space has the Radon-Nikodym property. And $W^{1,1}$ contains an isomorphic copy of $L^1$, it's essentially a direct sum of some copies of $L^1$. And $L^1$ fails the Radon-Nikodym property. –  Mar 03 '15 at 15:30
  • Are there any practical examples for such $v_k$ for given functionals? And how can I check whether this representation is unique? – MackeyTopology Dec 02 '23 at 21:21