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Concept
Compact operator
Summary
In functional analysis, a branch of mathematics, a compact operator is a linear operator , where are normed vector spaces, with the property that maps bounded subsets of to relatively compact subsets of (subsets with compact closure in ). Such an operator is necessarily a bounded operator, and so continuous. Some authors require that are Banach, but the definition can be extended to more general spaces.
Any bounded operator 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 is a Hilbert space, it is true that any compact operator is a limit of finite-rank operators, 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.
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.
A linear map between two topological vector spaces is said to be compact if there exists a neighborhood of the origin in such that is a relatively compact subset of .
Let be normed spaces and a linear operator. Then the following statements are equivalent, and some of them are used as the principal definition by different authors
is a compact operator;
the image of the unit ball of under is relatively compact in ;
the image of any bounded subset of under is relatively compact in ;
there exists a neighbourhood of the origin in and a compact subset such that ;
for any bounded sequence in , the sequence contains a converging subsequence.
Official source
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