Dual spaces

Continuous linear functionals on a normed space form its continuous dual.

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Compact operators
A compact operator maps bounded sets into relatively compact sets.
Hahn–Banach theorem
Appropriately bounded linear functionals extend from subspaces while preserving their bounds.
Normed spaces
A norm measures vector size with homogeneity, separation and the triangle inequality.
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.