I was asked to provide an example of an incomplete orthogonal in any $L^2$ space and was wondering if this is a valid answer.
Take some known complete system, for example the Hermite polynomials over the entire real line. Consider the same set but with some random member removed. The new set is still an ortogonal set, but you can't construct the removed member since it's orthogonal to the other members.
Is this a valid example?