Is $\overline{A \cap P} = \bar A \cap \bar P$ when $A$ is a closed convex set and $P$ is a convex cone?

Question

Let $A \subset L^2(\Omega)$ be a closed convex set and let $P:=\{ u \in L^2(\Omega) : u \geq 0 \text{ a.e}\}$.

Is it true that $\overline{A \cap P} = \bar A \cap \bar P$?

Here the bars denote the closure in $L^2(\Omega)$.

I know the left set is a subset of the right and that's all we get in general but this is a special case.. does it hold true ?

Answer 1

Some basic observations: $P$ is closed in $L^{2}(\Omega)$ so $A\cap P$ is also closed. So your question is whether $A\cap P=A \cap P$ which is of course true.