# On integrability in F-spaces

Studia Mathematica (1994)

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

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## Abstract

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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 ${x}^{\text{'}}\left({t}_{0}\right)={x}_{0}$ for given ${t}_{0}$ and ${x}_{0}$ and x’(t) = 0 for $t\ne {t}_{0}$.

## How to cite

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Popov, 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

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1. [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. [2] N. J. Kalton, Curves with zero derivatives in F-spaces, Glasgow Math. J. 22 (1981), 19-29. Zbl0454.46001
3. [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. [4] S. Mazur and W. Orlicz, Sur les espaces métriques linéaires I, Studia Math. 10 (1948), 184-208. Zbl0036.07801
5. [5] S. Rolewicz, Metric Linear Spaces, PWN, Warszawa, 1985.

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