A well known theorem about Hilbert spaces states that if $L$ is a closed subspace of Hilbert space $H$ then $H = L \oplus L^\perp$.
But can there be orthogonal decomposition of Hilbert space for some non-closed subspace $L$?
If the answer is "no" for Hilbert spaces then what about general inner-product spaces?
Obviously $L$ has to be infinite-dimensional to be non-closed. Unfortunately i'm pretty bad at infinite-dimensional examples so didn't succeed to find one yet. I'm not even sure that it exists. Probably since $L^\perp$ is always closed the existence of orthogonal decomposition somehow implies that $L$ has to be closed as well.
Any ideas will be highly appreciated. Thanks in advance.