Spaceability in Banach and quasi-Banach sequence spaces

Abstract

Let X be a Banach space. We prove that, for a large class of Banach or quasi-Banach spaces E of X-valued sequences, the sets E − q∈Γ ℓ q (X), where Γ is any subset of (0, ∞], and E − c 0 (X) contain closed infinite-dimensional subspaces of E (if non-empty, of course). This result is applied in several particular cases and it is also shown that the same technique can be used to improve a result on the existence of spaces formed by norm-attaining linear operators.

Lineability and Spaceability: A New Approach

“…It is well known that ℓ p \ 0<q<p ℓ q is c-spaceable (see [11]). In this section we characterize (α, c)-spaceability in this setting.…”

Maximal spaceability in sequence spaces

“…Many interesting results in the field have been discovered lately, mainly concerning lineability/spaceability in functions spaces [1,3,4,7,8,[18][19][20][21][22]27], sequence spaces [1,2,6,9,23], spaces of linear operators/homogeneous polynomials [10,11,23,28,31], and even spaces of holomorphic functions [24]. Nevertheless, maximal spaceability is rarely obtained.…”

Lineability and spaceability for the weak form of Peano’s theorem and vector-valued sequence spaces

“…Thus (4) implies that h r (x) = a r f r (x) for all x ∈ B c 0 (N). As N is arbitrary, we have h r (x) = a r f r (x) for every x ∈ c 0 .…”

“…Section 2). Thereafter, the mother vector technique is used to obtain maximal-spaceability in a framework more general than that where c-spaceability was obtained in [4] (cf. Section 3).…”