Spectral theorem

Suitable self-adjoint or normal operators admit spectral decompositions; the precise form depends on the setting.

Different ways to understand it

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Compact operators
A compact operator maps bounded sets into relatively compact sets.
Self-adjoint operators
For bounded Hilbert-space operators, self-adjointness means equality with the adjoint.
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.