I am given a minimisation problem of finding $p \in \mathbb{P}^2$:
$\vert\vert e^x - p\vert\vert_{H^1(\Omega)}^2=\inf_{q \in \mathbb{P^2}}\vert\vert e^x -q\vert\vert_{H^1(\Omega)}$.
The norm $H^1$ is defined as a standard $W^{1,2}$ norm.
I could solve the problem using the standard procedure of minismising a functional, but the condition $p \in \mathbb{P}^2$ really confuses me: I assume $\mathbb{P}$ is a space of all polynomials, but then using $\mathbb{P}^2$ doesn't make much sense for me here. Is something wrong with my understanding of the problem? How should I proceed with solving it?