grandes-ecoles 2020 Q18

grandes-ecoles · France · centrale-maths2__mp Continuous Probability Distributions and Random Variables Verification of Probability Measure or Inner Product Properties

For all $k \in \mathbb{N}^*$, we set $g_k(x) = \sqrt{2}\sin(k\pi x)$. We denote by $G = \operatorname{Vect}\left((g_k)_{k \in \mathbb{N}^*}\right)$ and $H = G^\perp$. We admit that, $$H \neq \{0\} \Longrightarrow \exists f \in H \text{ such that } \left\{ \begin{array}{l} \|f\| = 1 \\ \langle T(f), f \rangle = \sup_{h \in H, \|h\|=1} \langle T(h), h \rangle \end{array} \right.$$ Deduce that $H = \{0\}$.

For all $k \in \mathbb{N}^*$, we set $g_k(x) = \sqrt{2}\sin(k\pi x)$. We denote by $G = \operatorname{Vect}\left((g_k)_{k \in \mathbb{N}^*}\right)$ and $H = G^\perp$.
We admit that,
$$H \neq \{0\} \Longrightarrow \exists f \in H \text{ such that } \left\{ \begin{array}{l} \|f\| = 1 \\ \langle T(f), f \rangle = \sup_{h \in H, \|h\|=1} \langle T(h), h \rangle \end{array} \right.$$
Deduce that $H = \{0\}$.

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