Let R [ X ] {\displaystyle {}\mathbb {R} [X]} be the polynomial ring over R {\displaystyle {}\mathbb {R} } . We set P 0 = 1 {\displaystyle {}P_{0}=1} , and, for n ∈ N + {\displaystyle {}n\in \mathbb {N} _{+}} , we set
Show that P n {\displaystyle {}P_{n}} , n ∈ N {\displaystyle {}n\in \mathbb {N} } , is a basis of R [ X ] {\displaystyle {}\mathbb {R} [X]} .
![{\displaystyle {}\mathbb {R} [X]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52a170b822ae65cbe9d1cb122f493fcd88305b7f)
