Prove that every incomplete inner product space $H$ has a closed vector subspace $H_0$ such that$H \neq H_0 \oplus H_0^T$
It's known that for complete inner product spaces, that is, Hilbert spaces, the statement is false, because for every closed vector subspace the equality holds.
I tried to find in the proof where's completeness is required, but it doesn't help me.
I've found a similar question, but there's only an example, but I need a general fact.
