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.
The answer appears to be "Yes". I have found a proof but it is too long for me to write here . The proof is an admission exams to the french "École Polytechnique" here is a link to the exam X-ENS PSI 2006 .I will try to write the proof in english in my spare time and add it here as soon as possible .