2. An operator $T \in B(H)$ is said to be positive if
$(Tx, x) \ge 0$
for all $x \in H$, and we write $T \ge 0$. Prove the following:
i) $T \ge 0$ implies that $T$ is self-adjoint.
ii) If $S \ge 0$, $T \ge 0$, $\alpha \ge 0$, then $S + \alpha T \ge 0$.
iii) If $T \ge 0$ ans $S \in B(H)$, then $S^*TS \ge 0$.
iv) If $T \in B(H)$, then $T^*T \ge 0$.
v) If $T$ is an orthonormal projection, then $T \ge 0$.