Discontinuous linear map
In mathematics, linear maps form an important class of "simple" functions which preserve the algebraic structure of linear spaces and are often used as approximations to more general functions (see linear approximation). If the spaces involved are also topological spaces (that is, topological vector spaces), then it makes sense to ask whether all linear maps are continuous. It turns out that for maps defined on infinite-dimensional topological vector spaces (e.g., infinite-dimensional normed spaces), the answer is generally no: there exist discontinuous linear maps. If the domain of definition is complete, it is trickier; such maps can be proven to exist, but the proof relies on the axiom of choice and does not provide an explicit example.
A linear functional which is not continuousA linear map which is not continuousDiscontinuous linear functionDiscontinuous linear functionalDiscontinuous linear operatorGeneral existence theorem of discontinuous mapsLinear discontinuous mapLinear operator which is not continuousNon-continuous linear functional
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Discontinuous linear map
In mathematics, linear maps form an important class of "simple" functions which preserve the algebraic structure of linear spaces and are often used as approximations to more general functions (see linear approximation). If the spaces involved are also topological spaces (that is, topological vector spaces), then it makes sense to ask whether all linear maps are continuous. It turns out that for maps defined on infinite-dimensional topological vector spaces (e.g., infinite-dimensional normed spaces), the answer is generally no: there exist discontinuous linear maps. If the domain of definition is complete, it is trickier; such maps can be proven to exist, but the proof relies on the axiom of choice and does not provide an explicit example.
has abstract
In mathematics, linear maps fo ...... t provide an explicit example.
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Operator liniowy nieciągły – o ...... przekształcenie nieciągłe z w
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数学において、線型写像は線型空間の「単に」代数構造を保つ写像 ...... 理を必要とするため、証明から明示的な例を得ることはできない。
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page length (characters) of wiki page
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969,544,563
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date
May 2015
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reason
A general "constructive" proof ...... rast seems kind of hand-wavey.
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This is not a proof nor even c ...... d as an established principle.
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wikiPageUsesTemplate
comment
In mathematics, linear maps fo ...... t provide an explicit example.
@en
Operator liniowy nieciągły – o ...... cznym opisie fizyki kwantowej.
@pl
数学において、線型写像は線型空間の「単に」代数構造を保つ写像 ...... 理を必要とするため、証明から明示的な例を得ることはできない。
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label
Discontinuous linear map
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Operator liniowy nieciągły
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不連続線型写像
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