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A bounded linear operator $T:X\to Y$ between Banach spaces is a weakly compact operator exactly when its second adjoint takes values in the canonical copy of $Y$ inside $Y^{\ast \ast }$. The Goldstine theorem proves necessity; the Banach-Alaoglu theorem, continuity of the second adjoint, and the Mazur theorem prove sufficiency.

The closed unit ball of a normed vector space is the set of vectors of norm at most one. In a continuous dual space it is compact in the weak-star topology by the Banach-Alaoglu theorem.

Let $K$ be a compact Hausdorff space. A test subspace $H\subset C(K,\mathbb{R})$ containing a strictly positive function is sufficient for uniform convergence of positive linear operators on continuous functions if, for each $x\in K$, the only positive linear functional agreeing with point evaluation on $H$ is that evaluation itself. By the Riesz-Markov-Kakutani representation theorem, this is uniqueness of the positive measure with the prescribed moments. Boundedness from the positive test function and the Banach-Alaoglu theorem prove the assertion by compactness. When $1\in H$, nonnegative tests with a unique zero at each prescribed $x$ give a convenient sufficient condition.

For a normed vector space $X$, take $K=B_{X^{\ast }}$ with the weak-star topology. Banach-Alaoglu theorem makes it compact Hausdorff, and each displayed evaluation function is continuous. Its supremum norm is $\Vert x\Vert$ by the Hahn-Banach theorem, giving a linear isometric embedding into $C(K)$.