From 81c8e0267f4c99e830a261cc4f6a899120c98e14 Mon Sep 17 00:00:00 2001 From: Alexis BAYLET Date: Fri, 27 Mar 2026 15:02:54 +0100 Subject: [PATCH] Add proof section for the uniqueness of reproducing kernel Hilbert spaces in RKHS document --- RKHS.md | 12 ++++++++++++ 1 file changed, 12 insertions(+) diff --git a/RKHS.md b/RKHS.md index c5f8ea1..8726672 100644 --- a/RKHS.md +++ b/RKHS.md @@ -48,3 +48,15 @@ Soit $k$ un noyau défini positif sur $\mathcal{X}$. 2. **Unicité :** Il existe un unique espace de Hilbert $H_k$ tel que $k$ soit son noyau reproduisant. Cet espace possède les propriétés : * $\forall x \in \mathcal{X}, k(\cdot, x) \in H_k$ * $\forall f \in H_k, \forall x \in \mathcal{X}, f(x) = \langle f, k(\cdot, x) \rangle_{H_k}$ + + +### Proof + +$$ H_{0} = \left \{ f : \mathbb{X} \to \mathbb{R}, (\alpha_{1}, \dots, \alpha_{n}) \in \mathbb{R}^{n}, f(x) = \sum_{i=1}^{n} \alpha_{i} k(x, x_{i}) \right \} $$ + +Let $ g(x) = \sum_{i=1}^{m} \beta_{i} k(x, z_{i}) $ + +$$ +\langle f, g \rangle_{H_0} = \sum_{i=1}^{n} \sum_{j=1}^{m} \alpha_{i} \beta_{j} k(x_{i}, z_{j}) +$$ +