Lineability of non-differentiable Pettis primitives

Abstract

Abstract. Let X be an infinite-dimensional Banach space. In 1995, settling a long outstanding problem of Pettis, Dilworth and Girardi constructed an X-valued Pettis integrable function on [0,1] whose primitive is nowhere weakly differentiable. Using their technique and some new ideas we show that ND, the set of strongly measurable Pettis integrable functions with nowhere weakly differentiable primitives, is lineable, i.e., there is an infinite dimensional vector space whose nonzero vectors belong to ND. IntroductionThroughout this note X is an infinite dimensional Banach space. For X-valued functions there are essentially two distinct notions of integration: the Bochner integral and the Pettis integral. The latter one includes properly the Bochner integral and preserves some of its good properties, e.g, countable additivity and absolute continuity of the indefinite integral, and convergence theorems. For X-valued functions on [0, 1] the Bochner indefinite integral is almost everywhere differentiable. As Pettis himself pointed out [17], the same property is not enjoyed by the Pettis integral. An interesting problem left open in [17] was whether the indefinite Pettis integral of a strongly measurable Pettis integrable function f is almost everywhere weakly differentiable (that is, does there exist a set E ⊂ [0, 1] of full measure such that t 0 x * f is differentiable to x * f for each t ∈ E and for each x * in the dual of X). Dilworth and Girardi [5]. They exhibited that every infinite dimensional Banach-space admits a Pettis integrable function from [0, 1] into X whose primitive is nowhere weakly differentiable. Their proof is rather flexible and gives the impression that there are many such functions. How does one make such a statement precise? One possibility is to show that in the space of all strongly measurable function from [0, 1] into X, the set of Pettis integrable functions with nowhere weakly differentiable primitive is a dense G δ set. This was done by Popa [20] in 2000. The topology one uses in this setting is the topology generated by the Pettis norm. The shortcoming of this method is that the Pettis norm is not complete [4]. Hence, proving that the set of Pettis integrable functions with nowhere weakly differentiable primitive is a dense G δ set loses some of its significance.An alternate notion of bigness in Banach space was introduced by Gurarity [11] and followed up in. This notion of bigness is of algebraic nature. If X is a Banach space then a subset M of X is lineable if M ∪{0} contains an infinite dimensional vector space. If, moreover, this infinite dimensional vector space is closed in the norm topology, then M is said to be spaceable. During the last twenty years many classical, pathological subsets of Banach spaces have been shown to be lineable or spaceable. What is surprising is that most of these sets are far from being vector spaces. For example, Gurarity [12] showed that the space of continuous nowhere differentiable functions on [0, 1] is lineable. The spaceability of this set was ...

Undominated Sequences of Integrable Functions

“…There are in the literature a number of results related to the topic we are concerned with, see for instance [9,21]. Unless otherwise stated, the measure considered on an interval of R will be always the Lebesgue measure m.…”

Rolewicz-type chaotic operators

“…We note that Equality (10) follows from the fact that c([σ] k ) = c([τ ] k ) whenever σ ∼ k τ . Equality (11) follows from the definitions of c([σ] k ) and A(n, k). Hence we have just shown that for 1 ≤ k ≤ j, we have…”

“…For example, the set of nowhere differentiable functions is lineable [18]. For other examples and survey on the subject refer to [11] [13]. Clearly, the set of chaotic operators cannot be lineable as no operator with norm less than one is hypercyclic.…”

Rolewicz-type chaotic operators

“…Hence, in the case of a separable Banach space, a linear operator is hypercyclic if and only if it is transitive. The notion of chaos in the sense of Devaney [14] also applies in the setting of Banach spaces. We say that a bounded linear operator T : X → X is chaotic in the sense of Devaney if T is transitive and the set of periodic points of T is dense in X.…”

“…For example, the set of nowhere differentiable functions is lineable [19]. For other examples and survey on the subject refer to [8,9,13,15]. Clearly, the set of chaotic operators cannot be lineable as no operator with norm less than one is hypercyclic.…”