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- Compact_operator abstract "In functional analysis, a branch of mathematics, a compact operator is a linear operator L from a Banach space X to another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset of Y. Such an operator is necessarily a bounded operator, and so continuous. Any bounded operator L that has finite rank is a compact operator; indeed, the class of compact operators is a natural generalisation of the class of finite-rank operators in an infinite-dimensional setting. When Y 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 in the operator norm of the finite-rank operators. Whether this was true in general for Banach spaces (the approximation property) was an unsolved question for many years; in the end Per Enflo gave a counter-example. 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.".
- Compact_operator wikiPageID "577441".
- Compact_operator wikiPageRevisionID "599609271".
- Compact_operator hasPhotoCollection Compact_operator.
- Compact_operator subject Category:Compactness_(mathematics).
- Compact_operator subject Category:Operator_theory.
- Compact_operator comment "In functional analysis, a branch of mathematics, a compact operator is a linear operator L from a Banach space X to another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset of Y. Such an operator is necessarily a bounded operator, and so continuous.".
- Compact_operator label "Compact operator".
- Compact_operator label "Kompakter Operator".
- Compact_operator label "Operador compacto".
- Compact_operator label "Operator zwarty".
- Compact_operator label "Operatore compatto".
- Compact_operator label "Opérateur compact".
- Compact_operator label "Компактный оператор".
- Compact_operator label "コンパクト作用素".
- Compact_operator sameAs Kompakter_Operator.
- Compact_operator sameAs Opérateur_compact.
- Compact_operator sameAs Operatore_compatto.
- Compact_operator sameAs コンパクト作用素.
- Compact_operator sameAs 콤팩트_작용소.
- Compact_operator sameAs Operator_zwarty.
- Compact_operator sameAs Operador_compacto.
- Compact_operator sameAs m.02rvbg.
- Compact_operator sameAs Q1780743.
- Compact_operator sameAs Q1780743.
- Compact_operator wasDerivedFrom Compact_operator?oldid=599609271.
- Compact_operator isPrimaryTopicOf Compact_operator.