Non-complete inner product spaces with orthonormal basis

Question

It is known that every Hilbert space admits an orthonormal basis, but is it true that an inner product space which admits an orthonormal basis is necessarily complete as a metric space? Can you give a counter-example?

Answer 1

The space $l_0$ of sequences with at most finitely many non-zero terms (as a subspace of $l^{2}$) with the usual basis elements $(1,0,0...),(0,1,0...)...$ is incomplete.

Answer 2

In a Hilbert space, with orthonormal basis $\{e_n,n=1,2,3,\dots\}$, let $X$ be the set of finite linear combinations $\sum t_n e_n$. This will be an incomplete inner product space, and $e_n$ is still a complete orthonormal set.