Compact operator

Summary

A compact operator is a linear map between Banach spaces that sends bounded sets to relatively compact sets. Compact operators are the operators closest to finite-dimensional linear algebra: their nonzero spectrum consists of eigenvalues of finite multiplicity, which can accumulate only at $0$. They include the finite-rank operators and their norm limits, and integral operators with continuous kernels. They are the setting of the Fredholm alternative and of the spectral theorem for compact self-adjoint operators.