On integrability in F-spaces

• Autorzy: Mikhail M. Popov
• Języki publikacji: EN

Abstrakty

EN

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 $x'(t_0) = x_0$ for given $t_0$ and $x_0$ and x'(t) = 0 for $t ≠ t_0$.

Kategorie tematyczne

• 46A16: Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.)
• 46G05: Derivatives

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1994
• Tom: 110
• Numer: 3
• Strony: 205-220

Daty

wydano

1994

otrzymano

1993-03-22

poprawiono

1993-05-17

Twórcy

autor

Mikhail M. Popov

• Prospect L. Svobody 20, Kv. 230, 310202 Kharkov, Ukraine

Bibliografia

• [1] N. J. Kalton, The compact endomorphisms of $L_p (0 ≤ p ≤ 1)$, Indiana Univ. Math. J. 27 (1978), 353-381.
• [2] N. J. Kalton, Curves with zero derivatives in F-spaces, Glasgow Math. J. 22 (1981), 19-29.
• [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.
• [5] S. Rolewicz, Metric Linear Spaces, PWN, Warszawa, 1985.