Enhance RKHS document by adding Moore-Aronszajn theorem and clarifying kernel examples

RKHS.md

@@ -21,10 +21,22 @@ Let `H` be a Hilbert space of functions of real value functions $ f: \mathbb{X}
 
 **remark:** $ f = K(., x), k(x, x') = \langle k(., x'), k(., x) \rangle_{H} $
 
-## Examples of kernels
+**Examples of kernels:**
 
 ### kernel PDS
 
 - $k(x, x') = \exp(-\gamma \|x - x'\|^2) $ with $ x, x' \in \mathbb{X} $
 - $k(x, x') = (1 + \langle x, x' \rangle)^{p} $ with $ x, x' \in \mathbb{X} $
 
+## Moore Aronszajn Theorem (1943)
+
+Let `k` be a PDS kernel over $ \mathbb{X} $.
+
+There exists a Hilbert space `H` and a map $ \Phi: \mathbb{X} \to H $ such that $ \forall x, x' \in \mathbb{X}, k(x, x') = \langle \Phi(x), \Phi(x') \rangle_{H} $
+
+Moreover, there is a unique Hilbert space such that k is a reproducing kernel of `H`.
+
+Let's call `H` : `H_{k}`
+
+- $ \forall x \in \mathbb{X}, k(., x) \in H $
+- $ \forall f \in H, \forall x \in \mathbb{X}, f(x) = \langle f, k(., x) \rangle_{H} $