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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?

Moisej Braver
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    What is $\mathbf{P}^2$? I would guess that these are polynomials up to degree $2$? – gerw Oct 20 '17 at 06:24
  • I thought like that, but for example here $\mathbb{P}$ is used a space of all polynomials. Anyway, do you have any hints if $\mathbb{P}^2$ are polynomials up to degree 2? – Moisej Braver Oct 20 '17 at 07:59

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