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- Banach_limit abstract "In mathematical analysis, a Banach limit is a continuous linear functional defined on the Banach space of all bounded complex-valued sequences such that for any sequences and , the following conditions are satisfied: (linearity); if for all , then , where is the shift operator defined by . If is a convergent sequence, then . Hence, is an extension of the continuous functional In other words, a Banach limit extends the usual limits, is shift-invariant and positive. However, there exist sequences for which the values of two Banach limits do not agree. We say that the Banach limit is not uniquely determined in this case. However, as a consequence of the above properties, a Banach limit also satisfies: The existence of Banach limits is usually proved using the Hahn–Banach theorem (analyst's approach) or using ultrafilters (this approach is more frequent in set-theoretical expositions). These proofs use the Axiom of choice (so called non-effective proof).".
- Banach_limit wikiPageID "2591539".
- Banach_limit wikiPageRevisionID "594023067".
- Banach_limit hasPhotoCollection Banach_limit.
- Banach_limit id "7213".
- Banach_limit title "Banach limit".
- Banach_limit subject Category:Functional_analysis.
- Banach_limit comment "In mathematical analysis, a Banach limit is a continuous linear functional defined on the Banach space of all bounded complex-valued sequences such that for any sequences and , the following conditions are satisfied: (linearity); if for all , then , where is the shift operator defined by . If is a convergent sequence, then . Hence, is an extension of the continuous functional In other words, a Banach limit extends the usual limits, is shift-invariant and positive.".
- Banach_limit label "Banach limit".
- Banach_limit label "Banachlimes".
- Banach_limit label "Granica Banacha".
- Banach_limit label "Límite de Banach".
- Banach_limit label "Банаховы пределы".
- Banach_limit sameAs Banachlimes.
- Banach_limit sameAs Límite_de_Banach.
- Banach_limit sameAs Granica_Banacha.
- Banach_limit sameAs m.07q7wh.
- Banach_limit sameAs Q806060.
- Banach_limit sameAs Q806060.
- Banach_limit wasDerivedFrom Banach_limit?oldid=594023067.
- Banach_limit isPrimaryTopicOf Banach_limit.