On the Zeros and Closure of Orthogonal Functions

This note deals primarily with infinite sets of continuous functions Φ_n(x) uniformly bounded and orthonormal over (0,I), i.e. the sequence being Φ₁, Φ₂, ........., Φ_n,

∫Φ,Φ₁dx = δ_n, |Φ_n|≤M for all n, 0≤x≤I.