We use the notations of the previous parts. For $u \in \mathscr{M}_{d,1}(\mathbb{R})$, we denote $T(u) = (T_i(u))_{1 \leqslant i \leqslant d} \in \mathbb{R}^d$ the vector defined by $$T_i(u) = \operatorname{Var}\left(\langle L_i, u \rangle\right) \quad \text{for } i \in \{1,\ldots,d\}.$$ (a) Show that there exists $c_0 \geqslant 0$ such that for all $u \in \mathscr{M}_{d,1}(\mathbb{R})$, we have $\|T(u)\|_1 \leqslant c_0 \|u\|_2^2$. (b) Deduce the existence of $c_1 \geqslant 0$ such that for all $u \in \mathscr{M}_{d,1}(\mathbb{R})$, we have $\|T(u)\|_1 \leqslant c_1 \|u\|_1^2$.
We use the notations of the previous parts. For $u \in \mathscr{M}_{d,1}(\mathbb{R})$, we denote $T(u) = (T_i(u))_{1 \leqslant i \leqslant d} \in \mathbb{R}^d$ the vector defined by
$$T_i(u) = \operatorname{Var}\left(\langle L_i, u \rangle\right) \quad \text{for } i \in \{1,\ldots,d\}.$$
(a) Show that there exists $c_0 \geqslant 0$ such that for all $u \in \mathscr{M}_{d,1}(\mathbb{R})$, we have $\|T(u)\|_1 \leqslant c_0 \|u\|_2^2$.
(b) Deduce the existence of $c_1 \geqslant 0$ such that for all $u \in \mathscr{M}_{d,1}(\mathbb{R})$, we have $\|T(u)\|_1 \leqslant c_1 \|u\|_1^2$.