grandes-ecoles 2021 Q23

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 $\left( W _ { n } \right) _ { n \in \mathbb { N } }$ be another orthogonal system. Show that $\forall n \in \mathbb { N } , W _ { n } = V _ { n }$.

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 $\left( W _ { n } \right) _ { n \in \mathbb { N } }$ be another orthogonal system. Show that $\forall n \in \mathbb { N } , W _ { n } = V _ { n }$.

Paper Questions

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