Editing Compact operator (section)

{{Short description|Type of continuous linear operator}}
In [[functional analysis]], a branch of [[mathematics]], a '''compact operator''' is a [[linear operator]] <math>T: X \to Y</math>, where <math>X,Y</math> are [[normed vector space]]s, with the property that <math>T</math> maps [[Bounded set|bounded subsets]] of <math>X</math> to [[relatively compact]] subsets of <math>Y</math> (subsets with compact [[closure (topology)|closure]] in <math>Y</math>). Such an operator is necessarily a [[bounded operator]], and so continuous.<ref name="Conway 1985 loc=Section 2.4">{{harvnb|Conway|1985|loc=Section 2.4}}</ref> Some authors require that <math>X,Y</math> are [[Banach space|Banach]], but the definition can be extended to more general spaces.

Any bounded operator ''<math>T</math>'' that has finite [[rank of a linear operator|rank]] is a compact operator; indeed, the class of compact operators is a natural generalization of the class of [[finite-rank operator]]s in an infinite-dimensional setting. When ''<math>Y</math>'' is a [[Hilbert space]], it is true that any compact operator is a limit of finite-rank operators,<ref name="Conway 1985 loc=Section 2.4"/> so that the class of compact operators can be defined alternatively as the closure of the set of finite-rank operators in the [[norm topology]]. Whether this was true in general for Banach spaces (the [[approximation property]]) was an unsolved question for many years; in 1973 [[Per Enflo]] gave a counter-example, building on work by [[Alexander Grothendieck]] and [[Stefan Banach]].<ref>{{harvnb|Enflo|1973}}</ref>

The origin of the theory of compact operators is in the theory of [[integral equation]]s, where integral operators supply concrete examples of such operators. A typical [[Fredholm integral equation]] gives rise to a compact operator ''K'' on [[function space]]s; the compactness property is shown by [[equicontinuity]]. The method of approximation by finite-rank operators is basic in the numerical solution of such equations. The abstract idea of [[Fredholm operator]] is derived from this connection.