In $\mathbb { R } [ X ]$ equipped with an inner product $( \cdot \mid \cdot )$, let $\left( V _ { n } \right) _ { n \in \mathbb { N } }$ be an orthogonal system (a sequence of polynomials that is orthogonal and where each $V_n$ is monic of degree $n$). Show that, for all $n \in \mathbb { N }$, the family $\left( V _ { 0 } , V _ { 1 } , \ldots , V _ { n } \right)$ is an orthogonal basis of the vector space $\mathbb { R } _ { n } [ X ]$ of polynomials with real coefficients of degree at most $n$.
In $\mathbb { R } [ X ]$ equipped with an inner product $( \cdot \mid \cdot )$, let $\left( V _ { n } \right) _ { n \in \mathbb { N } }$ be an orthogonal system (a sequence of polynomials that is orthogonal and where each $V_n$ is monic of degree $n$). Show that, for all $n \in \mathbb { N }$, the family $\left( V _ { 0 } , V _ { 1 } , \ldots , V _ { n } \right)$ is an orthogonal basis of the vector space $\mathbb { R } _ { n } [ X ]$ of polynomials with real coefficients of degree at most $n$.