From an old exam problem set :
Given $f \in C^{\infty}_{0}(\mathbb{R})$, find $r \in \mathbb{R}$ such that $r = \int_{\mathbb{R}} f(x) dx$.
The task is to check if the above problem is :
a) Well-posed/ill-posed if $\|\cdot\|_{L^{1}(\mathbb{R})}$ is used on $C^{\infty}_{0}(\mathbb{R})$, and
b) Well-posed/ill-posed if $\|\cdot\|_{L^{\infty}(\mathbb{R})}$ is used on $C^{\infty}_{0}(\mathbb{R})$.
=> My intuition : I have a feeling that it is well-posed when $\|\cdot\|_{L^{1}(\mathbb{R})}$ is used and ill-posed when $\|\cdot\|_{L^{\infty}(\mathbb{R})}$ is used. I think somehow the continuous dependence will be the deciding factor. But I have no clue how to make my intuition rigorous (if the intuition is true at all). So, any proof / counter-example will be much appreciated.
Thanks in advance.
By continuous dependence, I presume you are asking when the map: $$ T : C^{\infty}_0(\Bbb R) \rightarrow \Bbb R $$ defined by sending $$ f \mapsto\int_{\Bbb R} f(x) \,\mathrm{d}x $$ is continuous with respect to the given norm. Here I'm assuming $C^{\infty}_0(\Bbb R) = C^{\infty}_c(\Bbb R) = $ compactly supported smooth functions to ensure the above is well-defined.
Since the above mapping is linear, it is bounded as a map $(C^{\infty}_c(\Bbb R),\lVert \cdot \rVert_{L^p(\Bbb R)}) \rightarrow \Bbb R$ if and only if there is a constant $C>0$ such that $$ |Tf| = \left| \int_{\Bbb R} f(x) \,\mathrm{d}x \right| \leq C \lVert f \rVert_{L^p(\Bbb R)} $$ for all $f \in C_0^{\infty}(\Bbb R).$
So for both case when $p=1$ and $p=\infty,$ you need to show that either such a $C$ exists independently of $f$ (dependence is continuous), or exhibit a sequence of $f_k \in C^{\infty}_c(\Bbb R)$ demonstrating that no such $C>0$ exists (problem is ill-posed). I assume you can take it from here?