On the existence of bounded continuous solution of Hammerstein integral equation
Azzeddine Bellour (2011)
Matematički Vesnik
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Azzeddine Bellour (2011)
Matematički Vesnik
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Noboru Endou, Keiko Narita, Yasunari Shidama (2008)
Formalized Mathematics
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In this article we prove the Fatou's Lemma and Lebesgue's Convergence Theorem [10].MML identifier: MESFUN10, version: 7.9.01 4.101.1015
Panchapagesan, T.V. (1995)
Divulgaciones Matemáticas
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Šremr, Jiří (2010)
Electronic Journal of Differential Equations (EJDE) [electronic only]
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Bárcenas, Diómedes (2000)
Divulgaciones Matemáticas
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B. Kirchheim, Tomasz Natkaniec (1992)
Fundamenta Mathematicae
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In [2] the question was considered in how many directions can a nonmeasurable plane set behave even "better" than the classical one constructed by Sierpiński in [6], in the sense that any line in a given direction intersects the set in at most one point. We considerably improve these results and give a much sharper estimate for the size of the sets of those "better" directions.
Haluška, Ján, Hutník, Ondrej (2010)
Banach Journal of Mathematical Analysis [electronic only]
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Keiko Narita, Noboru Endou, Yasunari Shidama (2008)
Formalized Mathematics
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In this article, we formalized the notion of the integral of a complex-valued function considered as a sum of its real and imaginary parts. Then we defined the measurability and integrability in this context, and proved the linearity and several other basic properties of complex-valued measurable functions. The set of properties showed in this paper is based on [15], where the case of real-valued measurable functions is considered.MML identifier: MESFUN6C, version: 7.9.01 4.101.1015 ...
Yasushige Watase, Noboru Endou, Yasunari Shidama (2010)
Formalized Mathematics
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This article is the continuation of [31]. We define the set of Lp integrable functions - the set of all partial functions whose absolute value raised to the p-th power is integrable. We show that Lp integrable functions form the Lp space. We also prove Minkowski's inequality, Hölder's inequality and that Lp space is Banach space ([15], [27]).
Jacek Hejduk (1997)
Colloquium Mathematicae
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Keiko Narita, Noboru Endou, Yasunari Shidama (2009)
Formalized Mathematics
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In this article, we formalized Lebesgue's Convergence theorem of complex-valued function. We proved Lebesgue's Convergence Theorem of realvalued function using the theorem of extensional real-valued function. Then applying the former theorem to real part and imaginary part of complex-valued functional sequences, we proved Lebesgue's Convergence Theorem of complex-valued function. We also defined partial sums of real-valued functional sequences and complex-valued functional sequences...