We fix two $p$-tuples $(x_i)_{i \in \llbracket 1,p \rrbracket}$ and $(a_i)_{i \in \llbracket 1,p \rrbracket}$ of real numbers with the $x_i$ pairwise distinct. We use the notation $\mathcal{S}$, $J$, $J_*$, $\mathcal{S}_*$, $(\mid)_{\mathcal{H}}$ as defined previously. Let $\mathcal{H}_0 = \{h \in \mathcal{H} \mid h(x_i) = 0\ \forall i \in \llbracket 1,p \rrbracket\}$ and $\tilde{h} \in \mathcal{S}_*$ (we assume here $\mathcal{S}_*$ non-empty). Show that $\left(\tilde{h} \mid h_0\right)_{\mathcal{H}} = 0$ for all $h_0 \in \mathcal{H}_0$.
We fix two $p$-tuples $(x_i)_{i \in \llbracket 1,p \rrbracket}$ and $(a_i)_{i \in \llbracket 1,p \rrbracket}$ of real numbers with the $x_i$ pairwise distinct. We use the notation $\mathcal{S}$, $J$, $J_*$, $\mathcal{S}_*$, $(\mid)_{\mathcal{H}}$ as defined previously.
Let $\mathcal{H}_0 = \{h \in \mathcal{H} \mid h(x_i) = 0\ \forall i \in \llbracket 1,p \rrbracket\}$ and $\tilde{h} \in \mathcal{S}_*$ (we assume here $\mathcal{S}_*$ non-empty).
Show that $\left(\tilde{h} \mid h_0\right)_{\mathcal{H}} = 0$ for all $h_0 \in \mathcal{H}_0$.