# On integrability in F-spaces

Studia Mathematica (1994)

- Volume: 110, Issue: 3, page 205-220
- ISSN: 0039-3223

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topPopov, Mikhail. "On integrability in F-spaces." Studia Mathematica 110.3 (1994): 205-220. <http://eudml.org/doc/216109>.

@article{Popov1994,

abstract = {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 $l_p$-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 $x^\{\prime \}(t_0) = x_0$ for given $t_0$ and $x_0$ and x’(t) = 0 for $t ≠ t_0$.},

author = {Popov, Mikhail},

journal = {Studia Mathematica},

keywords = {Riemann integral; -space; quasi-Banach spaces},

language = {eng},

number = {3},

pages = {205-220},

title = {On integrability in F-spaces},

url = {http://eudml.org/doc/216109},

volume = {110},

year = {1994},

}

TY - JOUR

AU - Popov, Mikhail

TI - On integrability in F-spaces

JO - Studia Mathematica

PY - 1994

VL - 110

IS - 3

SP - 205

EP - 220

AB - 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 $l_p$-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 $x^{\prime }(t_0) = x_0$ for given $t_0$ and $x_0$ and x’(t) = 0 for $t ≠ t_0$.

LA - eng

KW - Riemann integral; -space; quasi-Banach spaces

UR - http://eudml.org/doc/216109

ER -

## References

top- [1] N. J. Kalton, The compact endomorphisms of ${L}_{p}\left(0\le p\le 1\right)$, Indiana Univ. Math. J. 27 (1978), 353-381. Zbl0403.46032
- [2] N. J. Kalton, Curves with zero derivatives in F-spaces, Glasgow Math. J. 22 (1981), 19-29. Zbl0454.46001
- [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.
- [4] S. Mazur and W. Orlicz, Sur les espaces métriques linéaires I, Studia Math. 10 (1948), 184-208. Zbl0036.07801
- [5] S. Rolewicz, Metric Linear Spaces, PWN, Warszawa, 1985.

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