Proposition 8.14. in Brezis states that:$(W_0^{1,p} (Ω))^*=W^{-1,p^*} (Ω)$ and we have the representation:
$∀ F∈(W_0^{1,p} (Ω))^* ∃ f_0...f_n ∈L^{p^*} (Ω)$ such that $∀ u∈W_0^{1,p}(Ω)$ $F(u)=∫_Ω uf_0dx+∑_{i=1}^N∫_Ωf_i ∂_i udx$
Where:
$∂_i$ is the distribution of derivative with respect to the $i$ variable, and remark 21 states that this representation is a valid for $F∈(W^{1,p} (Ω))^*$ as well.
My question is regarding remark 20 on the same page. It states that F can be identified with the distribution $f_0-∑_{i=1}^n∂_i f_i$ without adding any clarification. So I was curious if this distribution representation can be useful. The idea is this:
the distribution $f_0-∑_{i=1}^n∂_i f_i$ can't act on $W_0^{1,p} (Ω)$ functions, but it can act and test functions so take a sequence of test functions $ϕ_n$ converging to $u$ in $W_0^{1,p} (Ω)$. We have that:
$F(u)=\lim_{n→∞} ∫_Ω ϕ_n f_0dx+∑_{i=1}^N∫_Ω f_i ∂_i ϕ_n dx.$ This is because F it's a continuous linear functional and $ϕ_n→u$ in $W_0^{1,p} (Ω)$. Now we'll using this fact and considering $⟨*,*⟩$ the dual bracket in $D^* (Ω)×D(Ω)$ we can do the following:
$F(u)=\lim_{n→∞} ∫_Ω ϕ_n f_0dx+∑_{i=1}^N∫_Ω f_i ∂_i ϕ_n dx=\lim_{n→∞}⟨f_0,ϕ_n⟩+∑_{i=1}^N⟨f_i,∂_i ϕ_n⟩=\lim_{n→∞}⟨f_0-∑_{i=1}^n∂_i f_i,ϕ_n⟩$
I am particularly interested in the last line, which gives a connection between F and its distribution representative. Please tell me if the above calculation are correct
