On integrability in F-spaces
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 $t ≠ t_0$.