Displaying similar documents to “Egoroff's Theorem”

The Measurability of Complex-Valued Functional Sequences

Keiko Narita, Noboru Endou, Yasunari Shidama (2009)

Formalized Mathematics

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In this article, we formalized the measurability of complex-valued functional sequences. First, we proved the measurability of the limits of real-valued functional sequences. Next, we defined complex-valued functional sequences dividing real part into imaginary part. Then using the former theorems, we proved the measurability of each part. Lastly, we proved the measurability of the limits of complex-valued functional sequences. We also showed several properties of complex-valued measurable...

The First Mean Value Theorem for Integrals

Keiko Narita, Noboru Endou, Yasunari Shidama (2008)

Formalized Mathematics

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In this article, we prove the first mean value theorem for integrals [16]. The formalization of various theorems about the properties of the Lebesgue integral is also presented.MML identifier: MESFUNC7, version: 7.8.09 4.97.1001

Lebesgue's Convergence Theorem of Complex-Valued Function

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...

More on Continuous Functions on Normed Linear Spaces

Hiroyuki Okazaki, Noboru Endou, Yasunari Shidama (2011)

Formalized Mathematics

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In this article we formalize the definition and some facts about continuous functions from R into normed linear spaces [14].

Fatou's Lemma and the Lebesgue's Convergence Theorem

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

More on the Continuity of Real Functions

Keiko Narita, Artur Kornilowicz, Yasunari Shidama (2011)

Formalized Mathematics

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In this article we demonstrate basic properties of the continuous functions from R to Rn which correspond to state space equations in control engineering.

The Lebesgue Monotone Convergence Theorem

Noboru Endou, Keiko Narita, Yasunari Shidama (2008)

Formalized Mathematics

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In this article we prove the Monotone Convergence Theorem [16].MML identifier: MESFUNC9, version: 7.8.10 4.100.1011

Sorting by Exchanging

Grzegorz Bancerek (2011)

Formalized Mathematics

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We show that exchanging of pairs in an array which are in incorrect order leads to sorted array. It justifies correctness of Bubble Sort, Insertion Sort, and Quicksort.

Integral of Complex-Valued Measurable Function

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 ...

Complex Integral

Masahiko Yamazaki, Hiroshi Yamazaki, Yasunari Shidama, Katsumi Wasaki (2009)

Formalized Mathematics

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In this article, we defined complex curve and complex integral. Then we have proved the linearity for the complex integral. Furthermore, we have proved complex integral of complex curve's connection is the sum of each complex integral of individual complex curve.

The Cauchy-Riemann Differential Equations of Complex Functions

Hiroshi Yamazaki, Yasunari Shidama, Yatsuka Nakamura, Chanapat Pacharapokin (2009)

Formalized Mathematics

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In this article we prove Cauchy-Riemann differential equations of complex functions. These theorems give necessary and sufficient condition for differentiable function.

Non-Leibniz algebras with logarithms do not have the trigonometric identity

D. Przeworska-Rolewicz (2000)

Banach Center Publications

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Let X be a Leibniz algebra with unit e, i.e. an algebra with a right invertible linear operator D satisfying the Leibniz condition: D(xy) = xDy + (Dx)y for x,y belonging to the domain of D. If logarithmic mappings exist in X, then cosine and sine elements C(x) and S(x) defined by means of antilogarithmic mappings satisfy the Trigonometric Identity, i.e. [ C ( x ) ] 2 + [ S ( x ) ] 2 = e whenever x belongs to the domain of these mappings. The following question arises: Do there exist non-Leibniz algebras with logarithms...