If we are given that $A:V \rightarrow V$ is an operator where $V$ is a real Hilbert space. If we are given that $A$ is bounded, strictly positive $\big(\langle Au,u \rangle > 0$ for all $u \neq 0\big)$ and a self-adjoint linear operator then does it also follow that $A$ positive definite $\big(\langle Au,u \rangle \geq \alpha\Vert u \Vert$ where $\alpha > 0\big)$?
Also, it’s a bit confusing as to which is the standard definition for a positive definite operator, since sometimes the definition that I gave for a strictly positive operator is also used for positive definite.
Question inspired by the beginning of the proof of the following theorem:
The link to the document is Document link
The Theorem is on page 40.