= Weak compactness of an operator and its adjoint
{title2=$T\text{ weakly compact}\Longleftrightarrow T^*\text{ weakly compact}$}
A <bounded linear operator> between <Banach spaces> is weakly compact exactly when its <Banach-space adjoint> is weakly compact. Equivalently, that adjoint is continuous from its domain's <weak-star topology> to its range's <weak topology>. For the reverse direction, the bidual condition applied to $T^*$ forces $T^{***}$ to annihilate every functional on $Y^{**}$ vanishing on $J_YY$; the <Hahn-Banach theorem> then places $T^{**}(X^{**})$ inside $J_YY$.
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