Compact operators

A compact operator maps bounded sets into relatively compact sets.

Different ways to understand it

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Dual spaces
Continuous linear functionals on a normed space form its continuous dual.
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.
Vector spaces
A vector space specifies addition and scalar multiplication satisfying the vector-space axioms over a chosen field.
Banach spaces
A Banach space is a normed vector space complete in its induced metric.