Self-adjoint operators

For bounded Hilbert-space operators, self-adjointness means equality with the adjoint.

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Operator spectrum
The spectrum consists of scalars where subtracting the scalar identity fails to have a bounded inverse.
Spectral theorem
Suitable self-adjoint or normal operators admit spectral decompositions; the precise form depends on the setting.
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.
Compact operators
A compact operator maps bounded sets into relatively compact sets.