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 $$\forall P \in \mathbb{R}_{2m}[X], \quad T(P) = P' \text{ and } M(P) = P^*$$ where $P^*(X) = P(-X)$. The map $S$ is defined by $S(v,w) = (v \mid T(w)) + (T(v) \mid w)$. Show that $$\forall (P,Q) \in E^2, \quad S(P,Q) = P(1)Q(1) - P(-1)Q(-1)$$
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
$$\forall P \in \mathbb{R}_{2m}[X], \quad T(P) = P' \text{ and } M(P) = P^*$$
where $P^*(X) = P(-X)$. The map $S$ is defined by $S(v,w) = (v \mid T(w)) + (T(v) \mid w)$.
Show that
$$\forall (P,Q) \in E^2, \quad S(P,Q) = P(1)Q(1) - P(-1)Q(-1)$$