# RKHS

## Kernel definition

Let `X` be a non empty set. Let $k: \mathbb{X} \times \mathbb{X} \to \mathbb{R}$ symetric and positive definite. 


$$
\forall (x_{1}, ..., x_{n}) \in \mathbb{X}^{n},
\forall c \in \mathbb{R}^{n},

\sum_{i=1}^{n} \sum_{j=1}^{n} c_{i} c_{j} k(x_{i}, x_{j}) \geq 0
$$

$ K_{i, j} = k(x_{i}, x_{j}) $

### 1.2 Reproducing Kernel

Let `H` be a Hilbert space of functions of real value functions $ f: \mathbb{X} \to \mathbb{R} $ endowed with the inner product $ \langle ., . \rangle_{H} $ k is a reproducing kernel if :

- $ \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} $

`H` is called the Reproducing Kernel Hilbert Space (RKHS) associated to `k`.


**remark:** $ f = K(., x) then k $