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. We assume that the $x_i$ are pairwise distinct. We denote $\mathcal{S} = \{h \in \mathcal{H} \mid h(x_i) = a_i\}$ the set of $h \in \mathcal{H}$ that equal $a_i$ at $x_i$ for all $i \in \llbracket 1,p \rrbracket$. We denote $J : \mathcal{H} \rightarrow \mathbf{R}$ defined by $J(h) = \frac{1}{2}\|h\|_{\mathcal{H}}^2$ and $J_* = \inf\{J(h) \mid h \in \mathcal{S}\}$. We denote $\mathcal{S}_* = \{h \in \mathcal{S} \mid J(h) = J_*\}$. Show that $\mathcal{S}_*$ has at most one element.
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. We assume that the $x_i$ are pairwise distinct. We denote $\mathcal{S} = \{h \in \mathcal{H} \mid h(x_i) = a_i\}$ the set of $h \in \mathcal{H}$ that equal $a_i$ at $x_i$ for all $i \in \llbracket 1,p \rrbracket$. We denote $J : \mathcal{H} \rightarrow \mathbf{R}$ defined by $J(h) = \frac{1}{2}\|h\|_{\mathcal{H}}^2$ and $J_* = \inf\{J(h) \mid h \in \mathcal{S}\}$. We denote $\mathcal{S}_* = \{h \in \mathcal{S} \mid J(h) = J_*\}$.
Show that $\mathcal{S}_*$ has at most one element.