Title: | Extensions of the representation theorems of Riesz and Fréchet (English) |

Author: | Prandini, J. C. |

Language: | English |

Journal: | Mathematica Bohemica |

ISSN: | 0862-7959 (print) |

ISSN: | 2464-7136 (online) |

Volume: | 118 |

Issue: | 3 |

Year: | 1993 |

Pages: | 297-312 |

Summary lang: | English |

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Category: | math |

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Summary: | We present two types of representation theorems: one for linear continuous operators on space of Banach valued regulated functions of several real variables and the other for bilinear continuous operators on cartesian products of spaces of regulated functions of a real variable taking values on Banach spaces. We use generalizations of the notions of functions of bounded variation in the sense of Vitali and Fréchet and the Riemann-Stieltjes-Dushnik or interior integral. A few applications using geometry of Banach spaces are given. (English) |

Keyword: | Riesz type representation theorem |

Keyword: | Fréchet type representation theorem |

Keyword: | representation theorems |

Keyword: | linear continuous operators on spaces of Banach valued regulated functions of several real variables |

Keyword: | bilinear continuous operators on cartesian products |

Keyword: | functions of bounded variation |

Keyword: | interior integral |

Keyword: | geometry of Banach spaces |

Keyword: | spaces of regulated functions of a real variable taking values in Banach spaces |

Keyword: | regulated functions |

MSC: | 26B30 |

MSC: | 46B99 |

MSC: | 46E15 |

MSC: | 46E40 |

MSC: | 46G10 |

MSC: | 47B38 |

idZBL: | Zbl 0799.46046 |

idMR: | MR1239124 |

DOI: | 10.21136/MB.1993.125924 |

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Date available: | 2009-09-24T21:00:23Z |

Last updated: | 2020-07-29 |

Stable URL: | http://hdl.handle.net/10338.dmlcz/125924 |

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Reference: | [1] C. S. Hönig: Volteгra-Stieltjes Integral Equations.Mathematical Studies 16, North Holland Pub. Comp., Amsterdam, 1975. |

Reference: | [2] M. Morse, W. Transue: A Calculus for Fréchet Variations.Journal of Indian Math. Soc. XIV (1950), 65-117. Zbl 0040.05801, MR 0039043 |

Reference: | [3] M. Morse, W. Transue: The Fréchet variation in the small, sector limits, and left decompositions.Canadian Journal of Math. 2 (1950), 344-374. MR 0037340, 10.4153/CJM-1950-033-1 |

Reference: | [4] G. C. Rocha-Filho: Integral de Riemann Vetorial e Geometria dos Espaços de Banach.doctoral thesis, IME-USP, 1979. |

Reference: | [5] J. A. Clarkson, C. R. Adams: On defìnitions of bounded variation for functions of two variables.Trans. Am. Math. Soc. 35 (1933), 824-854. MR 1501718, 10.1090/S0002-9947-1933-1501718-2 |

Reference: | [6] M. Fréchet: Sur les fonctionnelles bilineaires.Trans. Amer. Math. Soc. (1915), 215-234. MR 1501010, 10.2307/1988990 |

Reference: | [7] N. Dunford, J. T. Schwartz: Linear Operators.paгt I, p. 337, Interscience, 1967. |

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