grandes-ecoles 2021 Q22

grandes-ecoles · France · centrale-maths1__pc Sequences and Series Inner Product Spaces, Orthogonality, and Hilbert Space Structure on Sequence/Function Spaces

Let $\left( V _ { n } \right) _ { n \in \mathbb { N } }$ be an orthogonal system in $\mathbb { R } [ X ]$ equipped with an inner product $( \cdot \mid \cdot )$. Let $n \in \mathbb { N }$ and $P \in \mathbb { R } [ X ]$ such that $\operatorname { deg } P < n$. Show that $\left( V _ { n } \mid P \right) = 0$.

Let $\left( V _ { n } \right) _ { n \in \mathbb { N } }$ be an orthogonal system in $\mathbb { R } [ X ]$ equipped with an inner product $( \cdot \mid \cdot )$. Let $n \in \mathbb { N }$ and $P \in \mathbb { R } [ X ]$ such that $\operatorname { deg } P < n$. Show that $\left( V _ { n } \mid P \right) = 0$.

Paper Questions

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