Weak convergence

A sequence converges weakly when every continuous linear functional converges on it.

Different ways to understand it

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Hahn–Banach theorem
Appropriately bounded linear functionals extend from subspaces while preserving their bounds.
Riesz representation in Hilbert spaces
Each continuous linear functional on a Hilbert space is represented by an inner product with a unique vector.
Banach spaces
A Banach space is a normed vector space complete in its induced metric.
Bounded linear operators
A linear operator bounded by a constant times input norm is continuous.
Compact operators
A compact operator maps bounded sets into relatively compact sets.