Normed spaces

A norm measures vector size with homogeneity, separation and the triangle inequality.

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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.
Dual spaces
Continuous linear functionals on a normed space form its continuous dual.
Vector spaces
A vector space specifies addition and scalar multiplication satisfying the vector-space axioms over a chosen field.
Compact operators
A compact operator maps bounded sets into relatively compact sets.