Displaying similar documents to “On the non-existence of norms for some algebras of functions”

Topological Properties of Real Normed Space

Kazuhisa Nakasho, Yuichi Futa, Yasunari Shidama (2014)

Formalized Mathematics

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In this article, we formalize topological properties of real normed spaces. In the first part, open and closed, density, separability and sequence and its convergence are discussed. Then we argue properties of real normed subspace. Then we discuss linear functions between real normed speces. Several kinds of subspaces induced by linear functions such as kernel, image and inverse image are considered here. The fact that Lipschitz continuity operators preserve convergence of sequences...

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...

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.

Orthogonality in normed linear spaces: a classification of the different concepts and some open problems.

Carlos Benítez Rodríguez (1989)

Revista Matemática de la Universidad Complutense de Madrid

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Orthogonality in inner products is a binary relation that can be expressed in many ways without explicit mention to the inner product of the space. Great part of such definitions have also sense in normed linear spaces. This simple observation is at the base of many concepts of orthogonality in these more general structures. Various authors introduced such concepts over the last fifty years, although the origins of some of the most interesting results that can be obtained for these generalized...