Suppose \(\mathcal{P}\) denotes the space of polynomials on \(\mathbb{R}^d\). There exist an inner product on \(\mathcal{P}\), such that
\[ \langle p, \Delta q\rangle_{\mathcal{P}} = \langle x^2 p, q\rangle_{\mathcal{P}}. \]Suppose \(\mathcal{P}\) denotes the space of polynomials on \(\mathbb{R}^d\). There exist an inner product on \(\mathcal{P}\), such that
\[ \langle p, \Delta q\rangle_{\mathcal{P}} = \langle x^2 p, q\rangle_{\mathcal{P}}. \]