$H$ is real Hilbert space. $a\colon H\times H \to \mathbb R$ is a bilinear form on $H$ with $\lvert a(x,y)\rvert \leq C\lVert x\rVert \lVert y\rVert$ and $a(x,x) \geq \alpha \lVert x\rVert^2$. I would like to know if one of the following properties holds and how to prove it, may you could help me with that.
1) $T(H)$ is dense in $H$ where $T$ is a linear operator on $H$ such that $a(x,y)=\langle Tx,y\rangle$.
2) $T$ is injective and $T(H)$ is complete. It should be possible to prove it with $\lVert Tx\rVert \geq \alpha \lVert x\rVert$.
3) $T$ is an isomorphism from $H$ to $H$.