Displaying similar documents to “Farthest points and subdifferential in p -normed spaces.”

Bidual Spaces and Reflexivity of Real Normed Spaces

Keiko Narita, Noboru Endou, Yasunari Shidama (2014)

Formalized Mathematics

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In this article, we considered bidual spaces and reflexivity of real normed spaces. At first we proved some corollaries applying Hahn-Banach theorem and showed related theorems. In the second section, we proved the norm of dual spaces and defined the natural mapping, from real normed spaces to bidual spaces. We also proved some properties of this mapping. Next, we defined real normed space of R, real number spaces as real normed spaces and proved related theorems. We can regard linear...

On the non-existence of norms for some algebras of functions

Bertram Yood (1994)

Studia Mathematica

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Let C(Ω) be the algebra of all complex-valued continuous functions on a topological space Ω where C(Ω) contains unbounded functions. First it is shown that C(Ω) cannot have a Banach algebra norm. Then it is shown that, for certain Ω, C(Ω) cannot possess an (incomplete) normed algebra norm. In particular, this is so for Ω = ℝ n where ℝ is the reals.

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.

Separability of Real Normed Spaces and Its Basic Properties

Kazuhisa Nakasho, Noboru Endou (2015)

Formalized Mathematics

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In this article, the separability of real normed spaces and its properties are mainly formalized. In the first section, it is proved that a real normed subspace is separable if it is generated by a countable subset. We used here the fact that the rational numbers form a dense subset of the real numbers. In the second section, the basic properties of the separable normed spaces are discussed. It is applied to isomorphic spaces via bounded linear operators and double dual spaces. In the...

Baire's Category Theorem and Some Spaces Generated from Real Normed Space 1

Noboru Endou, Yasunari Shidama, Katsumasa Okamura (2006)

Formalized Mathematics

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As application of complete metric space, we proved a Baire's category theorem. Then we defined some spaces generated from real normed space and discussed each of them. In the second section, we showed the equivalence of convergence and the continuity of a function. In other sections, we showed some topological properties of two spaces, which are topological space and linear topological space generated from real normed space.