A compact operator is a type of linear operator that maps bounded sets to relatively compact sets in a Banach space. This means that when you apply a compact operator to a bounded sequence, the image of that sequence has a convergent subsequence. Compact operators are important in functional analysis and are often used in the study of differential equations and integral equations. One of the key properties of compact operators is that they can be approximated by finite-rank operators, which are simpler to analyze. Examples of compact operators include the integral operators defined on L^2 spaces and certain types of Hilbert spaces. Understanding compact operators helps in solving various mathematical problems and in the development of numerical methods.