Given a bilinear symmetric form $b(u,v)$ on a Hilbert space. I need to know some very basic facts. A reference where these are discussed would be greatly appreciated.
1) There exists a symmetric bounded linear operator $S$ such that $b(u,v)=\langle Su,v\rangle$
2) Also, I would like to know if it is true that the spectrum of a symmetric bounded linear operator is closed.
3) If 1 and 2 are true, it seems to me that there should not be a difference for a symmetric bounded bilinear form between being coercive or positive definite. Am I right?