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1
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Dual Spaces and Hahn-Banach Theorem

100%
Narita K., Endou N., Shidama Y.
Formalized Mathematics
|
2014
|
tom 22
|
nr 1
69-77
EN
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 the Hahn-Banach extension theorem in real normed spaces. We have used extensively used [17].
2
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Topological classification of strong duals to nuclear (LF)-spaces

70%
Banakh T.
Studia Mathematica
|
2000
|
tom 138
|
nr 3
201-208
EN
We show that the strong dual X' to an infinite-dimensional nuclear (LF)-space is homeomorphic to one of the spaces: $ℝ^ω$, $ℝ^∞$, $Q×ℝ^∞$, $ℝ^ω×ℝ^∞$, or $(ℝ^∞)^ω$, where $ℝ^∞ = lim ℝ^n$ and $Q=[-1,1]^ω$. In particular, the Schwartz space D' of distributions is homeomorphic to $(ℝ^∞)^ω$. As a by-product of the proof we deduce that each infinite-dimensional locally convex space which is a direct limit of metrizable compacta is homeomorphic either to $ℝ^∞$ or to $Q×ℝ^∞$. In particular, the strong dual to any metrizable infinite-dimensional Montel space is homeomorphic either to $ℝ^∞$ or to $Q×ℝ^∞$.
3
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Characterizations of Rotundity and Smoothness by Approximate Orthogonalities

70%
Stypuła T., Wójcik P.
Annales Mathematicae Silesianae
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2016
|
tom 30
|
nr 1
193-201
EN
In this paper we consider the approximate orthogonalities in real normed spaces. Using the notion of approximate orthogonalities in real normed spaces, we provide some new characterizations of rotundity and smoothness of dual spaces.
4
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Some non-homogeneous Hardy spaces on locally compact Vilenkin groups

61%
Lu S., Yang D.
Colloquium Mathematicum
|
1996
|
tom 69
|
nr 1
1-17
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