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?
Best Answer
1) There exist bijection between bounded bilinear operators and bounded opeartors. The proof of this fact requires Banach-Steinhaus theorem. If bilinear form is symmetric, then the respecitive opeartor is (obviously) symmetric too
2) Spectrum of any bounded operator is compact, and as the consequence closed
3) No, consider bilinear form $$ b:\ell_2\times\ell_2\to\mathbb{R}:(x,y)\mapsto\sum\limits_{i=1}^\infty n^{-1}x_iy_i $$ It is symmetric positive but not coercive.