I would like to show that for any incomplete inner product space $H$ there exists a closed subspace $H_0$ such that
$H_0 + H_0^{\bot}\neq H$
Here and there vere slightly relevant discussion.
I saw an example for continuous functions (complex inner product inherited from $L_2$) and a subspace of functions such that $\int_0^1 f(t) dt = \int_{-1}^{0} f(t) dt$ which is closed and is not the whole subspace, but the orthogonal complement is zero.
We could may be prove that there always exists a subspace such that the orthogonal complement to $H_0$ in the completion of the space $H$ doesn't lie in the image of $H$ under the standard isometric injection into the completion.
