It is shown that the order of Gateaux smoothness of bump functions on a wide class of Banach spaces with unconditional basis is not better than that of Fréchet differentiability. It is proved as well that in the separable case this order for Banach lattices satisfying a lower p-estimate for 1≤ p < 2 can be only slightly better.
Let be a bounded domain of Lyapunov and a holomorphic function in the cylinder and continuous on . If for each fixed in some set , with positive Lebesgue measure , the function of can be continued to a function holomorphic on the whole plane with the exception of some finite number (polar set) of singularities, then can be holomorphically continued to , where is some analytic (closed pluripolar) subset of .
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...