We place ourselves in the particular case where $E = \mathbb{R}_{2m}[X]$, with $m \geq 2$ a fixed natural integer. This vector space is equipped with the scalar product $$\forall (P,Q) \in E^2, \quad (P \mid Q) = \int_{-1}^{1} P(t)Q(t)\,dt$$ The two endomorphisms $T$ and $M$ of $E$ are defined by $T(P) = P'$ and $M(P) = P^*$ where $P^*(X) = P(-X)$. We set $$\mathbb{R}_k^0[X] = \{P \in \mathbb{R}_k[X] \mid P(-1) = 0 \text{ and } P(1) = 0\}$$ The subspace $G$ consists of elements $u \in E$ satisfying (a) $u \in \operatorname{Im}(T)$ and (b) $\forall v \in E, S(u,v) = 0$, where $S(P,Q) = P(1)Q(1) - P(-1)Q(-1)$. Determine the subspace $G$. Is hypothesis (H5) satisfied?
We place ourselves in the particular case where $E = \mathbb{R}_{2m}[X]$, with $m \geq 2$ a fixed natural integer. This vector space is equipped with the scalar product
$$\forall (P,Q) \in E^2, \quad (P \mid Q) = \int_{-1}^{1} P(t)Q(t)\,dt$$
The two endomorphisms $T$ and $M$ of $E$ are defined by $T(P) = P'$ and $M(P) = P^*$ where $P^*(X) = P(-X)$. We set
$$\mathbb{R}_k^0[X] = \{P \in \mathbb{R}_k[X] \mid P(-1) = 0 \text{ and } P(1) = 0\}$$
The subspace $G$ consists of elements $u \in E$ satisfying (a) $u \in \operatorname{Im}(T)$ and (b) $\forall v \in E, S(u,v) = 0$, where $S(P,Q) = P(1)Q(1) - P(-1)Q(-1)$.
Determine the subspace $G$. Is hypothesis (H5) satisfied?