## Topological structures in Colombeau algebras I: topological ˜ C-modules and (2004)

### BibTeX

@MISC{Theory04topologicalstructures,

author = {Duality Theory and Claudia Garetto},

title = {Topological structures in Colombeau algebras I: topological ˜ C-modules and},

year = {2004}

}

### OpenURL

### Abstract

We study modules over the ring ˜ C of complex generalized numbers from a topological point of view, introducing the notions of ˜ C-linear topology and locally convex ˜ C-linear topology. In this context particular attention is given to completeness, continuity of ˜ C-linear maps and elements of duality theory for topological ˜ C-modules. As main examples we consider various Colombeau algebras of generalized functions. Key words: modules over the ring of complex generalized numbers, algebras of generalized functions, topology, duality theory

### Citations

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(Show Context)
Citation Context ...4, 36, 37, 41]. In the recent research on the subject a variety of algebras of generalized functions [3, 12, 13, 17, 19, 22] have been introduced in addition to the original construction by Colombeau =-=[6, 7]-=- and investigated in its algebraic and structural aspects as well as in analytic and applicative aspects. These investigations have produced a theory of point values in the Colombeau algebra G(Ω) and ... |

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(Show Context)
Citation Context ...4, 36, 37, 41]. In the recent research on the subject a variety of algebras of generalized functions [3, 12, 13, 17, 19, 22] have been introduced in addition to the original construction by Colombeau =-=[6, 7]-=- and investigated in its algebraic and structural aspects as well as in analytic and applicative aspects. These investigations have produced a theory of point values in the Colombeau algebra G(Ω) and ... |

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Citation Context ... G is quasi-complete if every bounded closed subset is complete. Remark 1.29. A topological ˜ C-module G is a uniform space and hence the following properties hold which will be used repeatedly later =-=[5, 49]-=-. Let A be a subset of G. Any filter F on A convergent to some u ∈ G is a Cauchy filter and that if u ∈ G adheres to a Cauchy filter O on A then O converges to u. Moreover a complete subset of a separ... |

18 |
Generalized solutions to the Korteweg-De Vries and the regularized long-wave equations
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(Show Context)
Citation Context ...30, 13J99, 46A20 0 Introduction Colombeau algebras of generalized functions have proved to be an analytically powerful tool in dealing with linear and nonlinear PDEs with highly singular coefficients =-=[1, 2, 3, 8, 9, 10, 18, 21, 23, 24, 29, 30, 32, 33, 34, 36, 37, 41]-=-. In the recent research on the subject a variety of algebras of generalized functions [3, 12, 13, 17, 19, 22] have been introduced in addition to the original construction by Colombeau [6, 7] and inv... |

16 |
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(Show Context)
Citation Context ...30, 13J99, 46A20 0 Introduction Colombeau algebras of generalized functions have proved to be an analytically powerful tool in dealing with linear and nonlinear PDEs with highly singular coefficients =-=[1, 2, 3, 8, 9, 10, 18, 21, 23, 24, 29, 30, 32, 33, 34, 36, 37, 41]-=-. In the recent research on the subject a variety of algebras of generalized functions [3, 12, 13, 17, 19, 22] have been introduced in addition to the original construction by Colombeau [6, 7] and inv... |

8 |
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(Show Context)
Citation Context ...t class of examples of ˜ C-modules, whose general theory was developed in the previous sections. In the literature there already exist papers on topologies, generalized functions and applications cf. =-=[1, 4, 35]-=- which consider spaces of generalized functions GE based on a locally convex topological vector space E and define topologies in terms of valuations and ultra-pseudo-seminorms [32, 46, 47, 48, 50]. Th... |

5 |
A Nonlinear Theory of Generalized Functions. Number 1421
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(Show Context)
Citation Context ...30, 13J99, 46A20 0 Introduction Colombeau algebras of generalized functions have proved to be an analytically powerful tool in dealing with linear and nonlinear PDEs with highly singular coefficients =-=[1, 2, 3, 8, 9, 10, 18, 21, 23, 24, 29, 30, 32, 33, 34, 36, 37, 41]-=-. In the recent research on the subject a variety of algebras of generalized functions [3, 12, 13, 17, 19, 22] have been introduced in addition to the original construction by Colombeau [6, 7] and inv... |