Compact operator

In functional analysis, a branch of mathematics, a compact operator is a linear operator ${\displaystyle T:X\to Y}$, where ${\displaystyle X,Y}$ are normed vector spaces, with the property that ${\displaystyle T}$ maps bounded subsets of ${\displaystyle X}$ to relatively compact subsets of ${\displaystyle Y}$ (subsets with compact closure in ${\displaystyle Y}$). Such an operator is necessarily a bounded operator, and so continuous.^[1] Some authors require that ${\displaystyle X,Y}$ are Banach, but the definition can be extended to more general spaces.

Any bounded operator ${\displaystyle T}$ that has finite rank is a compact operator; indeed, the class of compact operators is a natural generalization of the class of finite-rank operators in an infinite-dimensional setting. When ${\displaystyle Y}$ is a Hilbert space, it is true that any compact operator is a limit of finite-rank operators,^[1] 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 Grothendieck and Banach.^[2]

The origin of the theory of compact operators is in the theory of integral equations, where integral operators supply concrete examples of such operators. A typical Fredholm integral equation gives rise to a compact operator K on function spaces; 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.