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 $\mathcal{S}$, $\mathcal{S}_*$, $\mathcal{H}_0$, $(\mid)_{\mathcal{H}}$, $\gamma_{2\lambda}$, $\tau_x$ as defined previously. We denote $\mathcal{H}_0^\perp = \{h \in \mathcal{H} \mid \forall h_0 \in \mathcal{H}_0\ (h \mid h_0)_{\mathcal{H}} = 0\}$ the orthogonal subspace to $\mathcal{H}_0$ in $\mathcal{H}$. (a) Show that $\mathcal{S}_* = \mathcal{S} \cap \mathcal{H}_0^\perp$. (b) Show that $\mathcal{H}_0^\perp$ contains the vector subspace of $\mathcal{H}$ spanned by the functions $\tau_{x_i}(\gamma_{2\lambda})$ for $i \in \llbracket 1,p \rrbracket$.
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 $\mathcal{S}$, $\mathcal{S}_*$, $\mathcal{H}_0$, $(\mid)_{\mathcal{H}}$, $\gamma_{2\lambda}$, $\tau_x$ as defined previously. We denote $\mathcal{H}_0^\perp = \{h \in \mathcal{H} \mid \forall h_0 \in \mathcal{H}_0\ (h \mid h_0)_{\mathcal{H}} = 0\}$ the orthogonal subspace to $\mathcal{H}_0$ in $\mathcal{H}$.
(a) Show that $\mathcal{S}_* = \mathcal{S} \cap \mathcal{H}_0^\perp$.
(b) Show that $\mathcal{H}_0^\perp$ contains the vector subspace of $\mathcal{H}$ spanned by the functions $\tau_{x_i}(\gamma_{2\lambda})$ for $i \in \llbracket 1,p \rrbracket$.