grandes-ecoles 2020 Q17

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 $H = G^\perp$. Justify that, for all $(f,g) \in E^2$, we have $$\langle T(f), g \rangle = \langle f, T(g) \rangle$$ One may use question 12.

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 $H = G^\perp$.
Justify that, for all $(f,g) \in E^2$, we have
$$\langle T(f), g \rangle = \langle f, T(g) \rangle$$
One may use question 12.

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