> [!definition] > > Let $\ch$ be a [[Hilbert Space|Hilbert space]] and $T \in L(\ch, \ch)$. $T$ is **coercive** if there exists $\eps > 0$ such that $T - \eps I \ge 0$ is [[Semipositive Operator|semipositive]]. > [!theorem] > > If $T$ is coercive, then $T$ is [[Bounded Self-Adjoint Operator|self-adjoint]] and a [[Space of Toplinear Isomorphisms|toplinear isomorphism]]. > > *Proof*. $T - \eps I \ge 0$ implies that $T \ge 0$, so $T$ is semipositive and self-adjoint. Since $\norm{Tx} \ge \eps \norm{x}$, $T$ is injective and has closed image. As $T = T^*$, the [[Dual Map|dual map]] $T^*$ is injective, meaning that the image of $T$ is dense. Due to the image also being closed, $T$ is injective and surjective. Lastly, $\norm{T^{-1}x} \le \norm{x}/\eps$, so $T^{-1}$ is bounded as well.