My apologies to moderator, but I think it would be better to go on in this topic in my next question.
It doesn't have anything with previous question, but I hope that's OK It goes like this:
If

is proper Hilbert's subspace of Hilbert's space

, then there exists non-zero vector who is orthogonal on

. Prove.
If we suppose otherwise that doesn't exists such

(

and

and

is normal to

), we have that

. From

and

we have that

.
But what then?

, so

and ... ?