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Rising Sun Lemma
In mathematical analysis, the rising sun lemma is a lemma due to Frigyes Riesz Frigyes Riesz ( hu, Riesz Frigyes, , sometimes spelled as Frederic; 22 January 1880 – 28 February 1956) was a HungarianEberhard Zeidler: Nonlinear Functional Analysis and Its Applications: Linear monotone operators. Springer, 199/ref> mathema ..., used in the proof of the Hardy–Littlewood maximal theorem. The lemma was a precursor in one dimension of the Calderón–Zygmund lemma. The lemma is stated as follows:See: * * * * :Suppose ''g'' is a real-valued continuous function on the interval 'a'',''b''and ''S'' is the set of ''x'' in 'a'',''b''such that there exists a ''y''∈(''x'',''b''] with ''g''(''y'') > ''g''(''x''). (Note that ''b'' cannot be in ''S'', though ''a'' may be.) Define ''E'' = ''S'' ∩ (''a'',''b''). :Then ''E'' is an open set, and it may be written as a countable union of disjoint intervals ::E=\bigcup_k (a_k,b_k) :such that ''g''(''a''''k'') = ''g''(''b''''k''), unle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rising Sun Lemma
In mathematical analysis, the rising sun lemma is a lemma due to Frigyes Riesz Frigyes Riesz ( hu, Riesz Frigyes, , sometimes spelled as Frederic; 22 January 1880 – 28 February 1956) was a HungarianEberhard Zeidler: Nonlinear Functional Analysis and Its Applications: Linear monotone operators. Springer, 199/ref> mathema ..., used in the proof of the Hardy–Littlewood maximal theorem. The lemma was a precursor in one dimension of the Calderón–Zygmund lemma. The lemma is stated as follows:See: * * * * :Suppose ''g'' is a real-valued continuous function on the interval 'a'',''b''and ''S'' is the set of ''x'' in 'a'',''b''such that there exists a ''y''∈(''x'',''b''] with ''g''(''y'') > ''g''(''x''). (Note that ''b'' cannot be in ''S'', though ''a'' may be.) Define ''E'' = ''S'' ∩ (''a'',''b''). :Then ''E'' is an open set, and it may be written as a countable union of disjoint intervals ::E=\bigcup_k (a_k,b_k) :such that ''g''(''a''''k'') = ''g''(''b''''k''), unle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (mathematics), series, and analytic functions. These theories are usually studied in the context of Real number, real and Complex number, complex numbers and Function (mathematics), functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any Space (mathematics), space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space). History Ancient Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lemma (mathematics)
In mathematics, informal logic and argument mapping, a lemma (plural lemmas or lemmata) is a generally minor, proven proposition which is used as a stepping stone to a larger result. For that reason, it is also known as a "helping theorem" or an "auxiliary theorem". In many cases, a lemma derives its importance from the theorem it aims to prove; however, a lemma can also turn out to be more important than originally thought. The word "lemma" derives from the Ancient Greek ("anything which is received", such as a gift, profit, or a bribe). Comparison with theorem There is no formal distinction between a lemma and a theorem, only one of intention (see Theorem terminology). However, a lemma can be considered a minor result whose sole purpose is to help prove a more substantial theorem – a step in the direction of proof. Well-known lemmas A good stepping stone can lead to many others. Some powerful results in mathematics are known as lemmas, first named for their originally min ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frigyes Riesz
Frigyes Riesz ( hu, Riesz Frigyes, , sometimes spelled as Frederic; 22 January 1880 – 28 February 1956) was a HungarianEberhard Zeidler: Nonlinear Functional Analysis and Its Applications: Linear monotone operators. Springer, 199/ref> mathematician who made fundamental contributions to functional analysis, as did his younger brother Marcel Riesz. Life and career He was born into a Jewish family in Győr, Austria-Hungary and died in Budapest, Hungary. Between 1911 and 1919 he was a professor at the Franz Joseph University in Kolozsvár, Austria-Hungary. The post-WW1 Treaty of Trianon transferred former Austro-Hungarian territory including Kolozsvár to the Kingdom of Romania, whereupon Kolozsvár's name changed to Cluj and the University of Kolozsvár moved to Szeged, Hungary, becoming the University of Szeged. Then, Riesz was the rector and a professor at the University of Szeged, as well as a member of the Hungarian Academy of Sciences. and the Polish Academy of Learning. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
