optimal state-value function

The optimal state-value function $v_{\ast }(s)$ is the maximum state-value function over all policies. In other words, $v_{\ast }(s)$ is the resulting state-value function when the agent follows the optimal policy (treat $\pi$ as a parameter in the equation below).

$$v_{\ast }(s)=\underset{\pi }{\max }{v}_{\pi }(s)$$