**($\Rightarrow$):** Suppose $E$ is reflexive, so the canonical embedding $\phi: E \to E^{**}$ is surjective. Then $\phi(B_E) = B_{E^{**}}$. The inverse $\phi^{-1}: (E^{**}, \sigma(E^{**}, E^*)) \to (E, \sigma(E, E^*))$ is continuous: by the universal property of initial [topologies](/page/Topology), it suffices to show $F \circ \phi^{-1}$ is $\sigma(E^{**}, E^*)$-continuous for each $F \in E^*$. But $(F \circ \phi^{-1})(g) = g(F)$ for $g \in E^{**}$, which is the evaluation map at $F$ — one of the generators of $\sigma(E^{**}, E^*)$. By the [Banach-Alaoglu Theorem](/theorems/212), $B_{E^{**}}$ is $\sigma(E^{**}, E^*)$-compact. Hence $B_E = \phi^{-1}(B_{E^{**}})$ is the continuous preimage of a compact [set](/page/Set), and is therefore $\sigma(E, E^*)$-compact. **($\Leftarrow$):** Suppose $B_E$ is weakly compact. The canonical embedding $\phi: (E, \sigma(E, E^*)) \to (E^{**}, \sigma(E^{**}, E^*))$ is continuous (composing $\phi$ with any $\sigma(E^{**}, E^{***})$-continuous functional yields a strongly continuous functional on $E$, hence an element of $E^*$, which is $\sigma(E, E^*)$-continuous by definition). Since $\sigma(E^{**}, E^*)$ is coarser than $\sigma(E^{**}, E^{***})$, [continuity](/page/Continuity) into the weak-$*$ topology follows. Hence $\phi(B_E)$ is $\sigma(E^{**}, E^*)$-compact, and in particular $\sigma(E^{**}, E^*)$-closed. By the [Goldstine Lemma](/theorems/898), $\phi(B_E)$ is $\sigma(E^{**}, E^*)$-dense in $B_{E^{**}}$. A dense subset that is also closed equals the whole set, so $\phi(B_E) = B_{E^{**}}$. Rescaling gives $\phi(E) = E^{**}$, so $E$ is reflexive.