Hilbert spaces

A Hilbert space is complete under a norm induced by an inner product.

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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.
Riesz representation in Hilbert spaces
Each continuous linear functional on a Hilbert space is represented by an inner product with a unique vector.
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.