We set
$$\forall ( P , Q ) \in \mathbb { R } [ X ] \times \mathbb { R } [ X ] , \quad ( P \mid Q ) = \frac { 2 } { \pi } \int _ { 0 } ^ { 1 } P ( 4 x ) Q ( 4 x ) \frac { \sqrt { 1 - x } } { \sqrt { x } } \mathrm { ~d} x .$$
Show that $( \cdot \mid \cdot )$ is an inner product on $\mathbb { R } [ X ]$.