Compact operator

A linear operator that maps bounded sets to sets with compact closure. In normed spaces, this is equivalent to mapping every bounded sequence to a sequence with a convergent subsequence.

Relatively compact set: A subset whose closure is compact. The images of bounded sets under a compact operator are relatively compact.
Fredholm alternative: A theorem relating solvability of a linear equation involving a compact perturbation of the identity to its adjoint equation. Compactness underlies the finite-dimensional-like alternative for equations involving identity minus a compact operator.
Bounded set: A subset of a metric space contained in some ball of finite radius. Compact operators are defined by how they transform every bounded set.
Identity operator: The linear operator that maps every vector to itself. On an infinite-dimensional normed space, the identity is bounded but not compact.
Sequential compactness: The property that every sequence in a space has a convergent subsequence with limit in that space. In normed spaces, compact operators are characterized by subsequence convergence of bounded sequences.
Riesz–Schauder theory: The spectral theory of compact operators, including discreteness of nonzero spectrum and finite-dimensional eigenspaces. It describes the restricted spectrum that distinguishes compact operators from general bounded operators.
Compact space: A topological space in which every open cover has a finite subcover. Compact closure in the image is the defining destination property.
Bounded linear operator: A linear map between normed spaces that sends bounded sets to bounded sets. Compact operators are bounded, but boundedness alone does not guarantee relatively compact images.
Finite-rank operator: A linear operator whose range is finite-dimensional. Finite-rank operators are compact because bounded subsets of finite-dimensional spaces have compact closure.
Eigenvalue: A scalar for which a linear operator has a nonzero vector that it scales by that scalar. Every nonzero spectral value of a compact operator is an eigenvalue.

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