In this part, $E_1$ denotes the vector space of functions $f : [0,1] \rightarrow \mathbb{R}$ continuous, of class $\mathcal{C}^1$ piecewise, and satisfying $f(0) = f(1) = 0$. We denote by $N$ the norm associated with the inner product $(f \mid g) = \int_0^1 f'(t) g'(t) \, \mathrm{d}t$. For all $(s,t) \in [0,1]^2$, $K(s,t) = k_s(t)$ where $$k_s(t) = \begin{cases} t(1-s) & \text{if } t < s \\ s(1-t) & \text{if } t \geqslant s. \end{cases}$$ Prove that the pre-Hilbert space $(E_1, (\cdot \mid \cdot))$ is a reproducing kernel Hilbert space and that its reproducing kernel is the application $K$ defined in the previous part.
In this part, $E_1$ denotes the vector space of functions $f : [0,1] \rightarrow \mathbb{R}$ continuous, of class $\mathcal{C}^1$ piecewise, and satisfying $f(0) = f(1) = 0$. We denote by $N$ the norm associated with the inner product $(f \mid g) = \int_0^1 f'(t) g'(t) \, \mathrm{d}t$.\\
For all $(s,t) \in [0,1]^2$, $K(s,t) = k_s(t)$ where
$$k_s(t) = \begin{cases} t(1-s) & \text{if } t < s \\ s(1-t) & \text{if } t \geqslant s. \end{cases}$$
Prove that the pre-Hilbert space $(E_1, (\cdot \mid \cdot))$ is a reproducing kernel Hilbert space and that its reproducing kernel is the application $K$ defined in the previous part.