Some usual and unusual properties of the Riemann integral for functions x : [a,b] → X where X is an F-space are investigated. In particular, a continuous integrable -valued function (0 < p < 1) with non-differentiable integral function is constructed. For some class of quasi-Banach spaces X it is proved that the set of all X-valued functions with zero derivative is dense in the space of all continuous functions, and for any two continuous functions x and y there is a sequence of differentiable functions which tends to x uniformly and for which the sequence of derivatives tends to y uniformly. There is also constructed a differentiable function x with for given and and x’(t) = 0 for .
@article{bwmeta1.element.bwnjournal-article-smv110i3p205bwm,
author = {Mikhail Popov},
title = {On integrability in F-spaces},
journal = {Studia Mathematica},
volume = {108},
year = {1994},
pages = {205-220},
zbl = {0803.46004},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv110i3p205bwm}
}
Popov, Mikhail. On integrability in F-spaces. Studia Mathematica, Tome 108 (1994) pp. 205-220. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv110i3p205bwm/
[00000] [1] N. J. Kalton, The compact endomorphisms of , Indiana Univ. Math. J. 27 (1978), 353-381. | Zbl 0403.46032
[00001] [2] N. J. Kalton, Curves with zero derivatives in F-spaces, Glasgow Math. J. 22 (1981), 19-29. | Zbl 0454.46001
[00002] [3] N. J. Kalton, N. T. Peck and J. W. Roberts, An F-space Sampler, London Math. Soc. Lecture Note Ser. 89, Cambridge Univ. Press, Cambridge, 1984.
[00003] [4] S. Mazur and W. Orlicz, Sur les espaces métriques linéaires I, Studia Math. 10 (1948), 184-208. | Zbl 0036.07801
[00004] [5] S. Rolewicz, Metric Linear Spaces, PWN, Warszawa, 1985.