Weak topology

The coarsest topology on a vector space that makes a specified family of linear functionals continuous. For a normed space, the weak topology is generated by its continuous linear functionals.

Weak convergence: Convergence in a topology generated by linear functionals, tested by convergence of each functional’s values. A sequence converges weakly when every functional defining this topology converges on it.
Norm topology: The topology on a normed vector space generated by its norm. The norm topology is generally finer, requiring norm convergence rather than convergence under each functional.
Banach–Alaoglu theorem: The theorem that the closed unit ball of a dual space is compact in the weak-star topology. It supplies compactness in a related topology when weak compactness is unavailable.
Hahn–Banach theorem: A theorem extending bounded linear functionals while preserving their norm, with geometric separation consequences. Its separation results show how continuous linear functionals distinguish points and convex sets.
Riemann–Lebesgue lemma: The theorem that Fourier coefficients of an integrable function tend to zero at high frequencies. It yields weak convergence to zero for oscillatory exponentials in suitable function spaces.
Initial topology: The coarsest topology on a set that makes a specified family of maps continuous. The weak topology is the initial topology induced by its chosen linear functionals.
Weak-star topology: The topology on a dual space of pointwise convergence on the original vector space. It tests dual-space functionals by evaluation on vectors, unlike the weak topology’s testing by the full dual.
Reflexive space: A normed space whose canonical map onto its double dual is surjective. In reflexive Banach spaces, closed bounded sets are relatively weakly compact.
Locally convex topology: A vector-space topology generated by a family of seminorms and having convex neighborhoods of zero. Weak topologies are prominent examples within this broader class of vector-space topologies.
Orthonormal sequence: A sequence of unit vectors that are pairwise orthogonal in an inner-product space. In an infinite-dimensional Hilbert space, every orthonormal sequence converges weakly to zero but not in norm.

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