We were given the following example in our lecture:
Let $\mathbb{K}= \mathbb{R}$, $\Omega \subseteq \mathbb{R}^n$. For all $F\in L^2(\Omega)^n$, $f_0 \in L^2(\Omega)$, the following identity $$ <f, \varphi>= - \int_{\Omega} F(x) \cdot \nabla \varphi(x)dx + \int_{\Omega}f_0(x) \varphi(x)dx $$ for $\varphi \in H^1_0(\Omega)=W_{2,0}^1(\Omega)$ defines a functional in $H^{-1}(\Omega)=(H^1_0(\Omega))^*$ (=the dual space). Question: What is meant by "functional" (I thought of simply a linear map in there) and how can I show this?
It continues: As $F$ is only in $L^2(\Omega)=H^0(\Omega)=W_2^0(\Omega)$, $div(F)$ is in general no regular distribution but we have $div(F) \in H^{-1}(\Omega)$. Question: How do I show that $div(F)$ is no regular distribution? I already don't understand why it is a distribution at all? In this context, I also fail to see why the second part holds ($div(F) \in H^{-1}(\Omega)$)?
Thank you for your explanations.
A functional is a linear and continuous function from a normed vector space to $\mathbb K$. In this case, $f$ is a functional if and only if $f\in H^{-1}(\Omega)$.
So you want to show that $f$ defined via $\varphi\mapsto \langle f,\varphi \rangle$ is continuous (linear is obvious).
What you have to do (i will not provide every detail) is to show that $$|\langle f,\varphi\rangle | \leq C \| \varphi \|_{H_0^1(\Omega)}$$ for some constant $C$. The estimate for the second summand can be shown by poincare, and for the first you have to use some triangle inequalities and Hoelder.