Let $Y$ be an inner-product space, and let $A$ be an orthonormal system. We're trying to find a case to demonstrate the fact that even if for any given $x$ in $Y$ there's some $u$ in $A$ such that $\langle u,x \rangle \neq 0$, that doesn't imply that the system is complete.
We've asserted that $A$ should not be a subset of a complete orthonormal system, and thus could not be completed to one. That led us to the realization that it's essential that Gram-Schmidt's operation could not be applied, which probably means we need to find an IPS whose dimension is at least the continuum cardinality.
However, we were not able to actually construct such an example, any tips will be welcomed...
Rephrase: My wording seems to have caused some confusion, so I'll try again: Find an IPS Y and an orthogonal system $A\subset Y$ such that $\forall x\in Y \exists u\in A: \langle x,u\rangle \ne0$ yet A is not a complete system (e.g. not an orthonormal base, e.g. it's span is not dense in Y, e.g. It doesn't always conform to Parseval's equality. These are all equivalent for any IPS).