# Density of the roots of orthogonal polynomials

Consider the vector space of continuous real valued functions on a finite interval and the inner product defined by the integral over the interval . If we have a family of orthogonal polynomials such that their span is dense then each polynomial has exactly n distinct roots . I was wondering if these roots might be dense in the interval because i tried to think of these polynomials as interpolation polynomials.

• Are you assuming these polynomials form a dense basis? Why are you assuming the roots are distinct? Feb 14, 2017 at 17:59
• Feb 14, 2017 at 18:10
• Sorry i forgot to mention that my bad , yes i am assuming the span of this set is dense ( the term in french is "Famille totale" i can't find the english equivalent ) Feb 14, 2017 at 19:28