Banach–Alaoglu theorem

The Banach–Alaoglu theorem states that the closed unit ball of a normed space’s dual is compact in the weak-* topology.

Weak-* topology: The topology on a dual space defined by pointwise convergence on the original space. It is the topology in which the theorem’s dual unit ball is compact.
Dual space: The vector space of continuous linear functionals on a given normed space. The theorem concerns the unit ball inside this space of functionals.
Weak-* compactness: Compactness with respect to the weak-* topology on a dual space. The theorem provides this property for every dual unit ball.
Heine–Borel theorem: In finite-dimensional Euclidean space, a set is compact exactly when it is closed and bounded. Banach–Alaoglu gives compactness from boundedness in a weaker topology, even in infinite dimensions.
Tychonoff theorem: The product of any family of compact topological spaces is compact in the product topology. It supplies compactness for the product space containing the dual unit ball.
Continuous linear functional: A linear map between normed spaces that is continuous in their norm topologies. Elements of the dual are precisely the functionals whose ball the theorem studies.
Riesz representation theorem: Theorems identifying certain continuous linear functionals with measures or inner-product vectors. Compactness of bounded functional sets helps establish representation and existence results.
Norm topology: The topology generated by open balls in a normed space. Dual unit balls are generally not compact in this stronger topology.
Evaluation map: A map that sends a functional to its values on specified vectors. Evaluations embed dual functionals into a product whose coordinates are scalar values.
Operator norm: The norm of a bounded linear operator, defined by its largest action on the unit ball. It defines the closed unit ball in the dual space.

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