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Displaying similar documents to “Riemann Integral of Functions from R into Real Normed Space”

Banach Algebra of Bounded Complex-Valued Functionals

Katuhiko Kanazashi, Hiroyuki Okazaki, Yasunari Shidama (2011)

Formalized Mathematics

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In this article, we describe some basic properties of the Banach algebra which is constructed from all bounded complex-valued functionals.

Functional Space C (ω), C 0 (ω)

Katuhiko Kanazashi, Hiroyuki Okazaki, Yasunari Shidama (2012)

Formalized Mathematics

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In this article, first we give a definition of a functional space which is constructed from all complex-valued continuous functions defined on a compact topological space. We prove that this functional space is a Banach algebra. Next, we give a definition of a function space which is constructed from all complex-valued continuous functions with bounded support. We also prove that this function space is a complex normed space.

Contracting Mapping on Normed Linear Space

Keiichi Miyajima, Artur Korniłowicz, Yasunari Shidama (2012)

Formalized Mathematics

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In this article, we described the contracting mapping on normed linear space. Furthermore, we applied that mapping to ordinary differential equations on real normed space. Our method is based on the one presented by Schwarz [29].

Dual Spaces and Hahn-Banach Theorem

Keiko Narita, Noboru Endou, Yasunari Shidama (2014)

Formalized Mathematics

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In this article, we deal with dual spaces and the Hahn-Banach Theorem. At the first, we defined dual spaces of real linear spaces and proved related basic properties. Next, we defined dual spaces of real normed spaces. We formed the definitions based on dual spaces of real linear spaces. In addition, we proved properties of the norm about elements of dual spaces. For the proof we referred to descriptions in the article [21]. Finally, applying theorems of the second section, we proved...

Banach’s Continuous Inverse Theorem and Closed Graph Theorem

Hideki Sakurai, Hiroyuki Okazaki, Yasunari Shidama (2012)

Formalized Mathematics

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In this article we formalize one of the most important theorems of linear operator theory - the Closed Graph Theorem commonly used in a standard text book such as [10] in Chapter 24.3. It states that a surjective closed linear operator between Banach spaces is bounded.

Differentiable Functions on Normed Linear Spaces

Yasunari Shidama (2012)

Formalized Mathematics

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In this article, we formalize differentiability of functions on normed linear spaces. Partial derivative, mean value theorem for vector-valued functions, continuous differentiability, etc. are formalized. As it is well known, there is no exact analog of the mean value theorem for vector-valued functions. However a certain type of generalization of the mean value theorem for vector-valued functions is obtained as follows: If ||ƒ'(x + t · h)|| is bounded for t between 0 and 1 by some constant...