Hahn–Banach theorem

Appropriately bounded linear functionals extend from subspaces while preserving their bounds.

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Dual spaces
Continuous linear functionals on a normed space form its continuous dual.
Weak convergence
A sequence converges weakly when every continuous linear functional converges on it.
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.