Bidual Spaces and Reflexivity of Real Normed Spaces

Formalized Mathematics (2014)

• Volume: 22, Issue: 4, page 303-311
• ISSN: 1426-2630

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Abstract

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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 functionals as linear operators by this definition. Accordingly we proved Uniform Boundedness Theorem for linear functionals using the theorem (5) from [21]. Finally, we defined reflexivity of real normed spaces and proved some theorems about isomorphism of linear operators. Using them, we proved some properties about reflexivity. These formalizations are based on [19], [20], [8] and [1].

How to cite

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Keiko Narita, Noboru Endou, and Yasunari Shidama. "Bidual Spaces and Reflexivity of Real Normed Spaces." Formalized Mathematics 22.4 (2014): 303-311. <http://eudml.org/doc/270906>.

@article{KeikoNarita2014,
abstract = {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 functionals as linear operators by this definition. Accordingly we proved Uniform Boundedness Theorem for linear functionals using the theorem (5) from [21]. Finally, we defined reflexivity of real normed spaces and proved some theorems about isomorphism of linear operators. Using them, we proved some properties about reflexivity. These formalizations are based on [19], [20], [8] and [1].},
author = {Keiko Narita, Noboru Endou, Yasunari Shidama},
journal = {Formalized Mathematics},
keywords = {continuous dual space; topological duality; reflexivity},
language = {eng},
number = {4},
pages = {303-311},
title = {Bidual Spaces and Reflexivity of Real Normed Spaces},
url = {http://eudml.org/doc/270906},
volume = {22},
year = {2014},
}

TY - JOUR
AU - Keiko Narita
AU - Noboru Endou
AU - Yasunari Shidama
TI - Bidual Spaces and Reflexivity of Real Normed Spaces
JO - Formalized Mathematics
PY - 2014
VL - 22
IS - 4
SP - 303
EP - 311
AB - 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 functionals as linear operators by this definition. Accordingly we proved Uniform Boundedness Theorem for linear functionals using the theorem (5) from [21]. Finally, we defined reflexivity of real normed spaces and proved some theorems about isomorphism of linear operators. Using them, we proved some properties about reflexivity. These formalizations are based on [19], [20], [8] and [1].
LA - eng
KW - continuous dual space; topological duality; reflexivity
UR - http://eudml.org/doc/270906
ER -

References

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