Banach spaces

A Banach space is a normed vector space complete in its induced metric.

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Hilbert spaces
A Hilbert space is complete under a norm induced by an inner product.
Normed spaces
A norm measures vector size with homogeneity, separation and the triangle inequality.
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.
Dual spaces
Continuous linear functionals on a normed space form its continuous dual.