On normed semialgebras
S. Bourne (1961)
Studia Mathematica
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S. Bourne (1961)
Studia Mathematica
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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...
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 where ℝ is the reals.
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.
Alexander Pruss (1995)
Studia Mathematica
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Let X be any topological space, and let C(X) be the algebra of all continuous complex-valued functions on X. We prove a conjecture of Yood (1994) to the effect that if there exists an unbounded element of C(X) then C(X) cannot be made into a normed algebra.
Putcha, Mohan.S., Yaqub, Adil (1979)
International Journal of Mathematics and Mathematical Sciences
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S. Bourne (1962)
Studia Mathematica
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Ursul, M.I. (2001)
Beiträge zur Algebra und Geometrie
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Pap, Endre (1996)
Novi Sad Journal of Mathematics
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Ali H. Handam, Hani A. Khashan (2017)
Open Mathematics
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An element in a ring R with identity is said to be strongly nil clean if it is the sum of an idempotent and a nilpotent that commute, R is said to be strongly nil clean if every element of R is strongly nil clean. Let C(R) be the center of a ring R and g(x) be a fixed polynomial in C(R)[x]. Then R is said to be strongly g(x)-nil clean if every element in R is a sum of a nilpotent and a root of g(x) that commute. In this paper, we give some relations between strongly nil clean rings and...
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...