The spaces of functions of finite upper -variation

Author:
Robert R. Nelson

Journal:
Trans. Amer. Math. Soc. **253** (1979), 171-190

MSC:
Primary 46E30; Secondary 28B05

MathSciNet review:
536941

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Abstract: Let *y* be a Banach space, , and be the semi-normed space of *Y*-valued Bochner measurable functions of a real variable which have finite upper *p*-variation. Let be the space of -equivalence classes. An averaging operator is defined with the aid of the theory of helixes in Banach spaces, which enables us to show that the spaces are Banach spaces, to characterize their members, and to show that they are isometrically isomorphic to Banach spaces of *Y*-valued measures with bounded *p*-variation.

**[1]**R. P. Boas Jr.,*Functions which are odd about several points*, Nieuw Arch. Wiskunde (3)**1**(1953), 27–32. MR**0054836****[2]**Paul L. Butzer and Hubert Berens,*Semi-groups of operators and approximation*, Die Grundlehren der mathematischen Wissenschaften, Band 145, Springer-Verlag New York Inc., New York, 1967. MR**0230022****[3]**J. Diestel and J. J. Uhl Jr.,*The Radon-Nikodym theorem for Banach space valued measures*, Rocky Mountain J. Math.**6**(1976), no. 1, 1–46. MR**0399852****[4]**N. Dunford and J. T. Schwartz,*Linear operators*. I, Interscience, New York, 1958.**[5]**G. H. Hardy and J. E. Littlewood,*Some properties of fractional integrals. I*, Math. Z.**27**(1928), no. 1, 565–606. MR**1544927**, 10.1007/BF01171116**[6]**Einar Hille and Ralph S. Phillips,*Functional analysis and semi-groups*, American Mathematical Society Colloquium Publications, vol. 31, American Mathematical Society, Providence, R. I., 1957. rev. ed. MR**0089373****[7]**Solomon Leader,*The theory of 𝐿^{𝑝}-spaces for finitely additive set functions*, Ann. of Math. (2)**58**(1953), 528–543. MR**0058126****[8]**J. K. Lee,*The completeness of the class of functions of bounded upper p-variation*, , Notices Amer. Math. Soc.**17**(1970), 1057. Abstract #681-B5.**[9]**P. Masani,*On helixes in Banach spaces*, Sankhyā Ser. A**38**(1976), no. 1, 1–27. MR**0471051****[10]**P. Masani,*Measurability and Pettis integration in Hilbert spaces*, J. Reine Angew. Math.**297**(1978), 92–135. MR**0463394****[11]**J. J. Uhl Jr.,*Orlicz spaces of finitely additive set functions*, Studia Math.**29**(1967), 19–58. MR**0226395**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1979-0536941-8

Keywords:
Upper *p*-variation,
average vector,
helixes in Banach spaces,
translation operators,
Lebesgue-Bochner integral,
chordal length function,
absolutely continuous function,
bounded variation

Article copyright:
© Copyright 1979
American Mathematical Society