Here are a few more consequences:
- The norm of $T^*$ equals the norm of $T$
- $X$ reflexive and $M$ closed in $X$ implies $M$ reflexive
- If $X$ Banach then $X^*$ reflexive iff $X$ is reflexive
- $X^*$ separable implies $X$ separable
- If $X$ Banach and $T \in B(X)$ then $T$ invertible iff $T^*$ invertible
