Dense lineability and algebrability of ℓ^{∞}∖𝑐₀

Abstract

We show that the set ℓ ∞ ∖ c 0 \ell ^{\infty }\setminus c_0 is maximal dense-lineable and densely strongly c \mathfrak {c} -algebrable answering a question posed by Nestoridis and complementing a result by García-Pacheco, Martín and Seoane-Sepúlveda.

Dense lineability and spaceability in certain subsets of $\ell_{\infty}$

“…In the first part of this section we prove that L(ω) is densely lineable in ℓ ∞ , thus complementing the result from [21] that L(c) is densely lineable in ℓ ∞ . Then we pass to the main part of the section, which is dedicated to the proof, under Martin's Axiom, that 2 n<ω L(n) is densely lineable in ℓ ∞ .…”

“…Moreover, for separable X, these conditions are equivalent to X \ Y being densely lineable in X [5]. For non-separable spaces, Papathanasiou [21] very recently proved that ℓ ∞ \c 0 is densely lineable in ℓ ∞ . It is however most unfortunate that his result is actually consequence of [5]; indeed, the very same proof of [5,Theorem 2.5] gives the complete characterisation that X \ Y is densely lineable in X if and only if dim(X/Y ) dens(X).…”

“…Yet, inspection of the proof in [21] gives the following interesting result: there is a dense subspace V of ℓ ∞ such that every non-zero vector in V has exactly continuum many accumulation points (see Remark 3.3). This result was the starting point of our research, as we were pondering lineability results for subsets of ℓ ∞ with a prescribed number of accumulation points (see [3] for some results in a similar direction).…”

“…In this notation, the result in [21] asserts that L(c) is densely lineable in ℓ ∞ . Similarly, our first result asserts that L(ω) is densely lineable in ℓ ∞ (Theorem 3.2).…”

Uncountably infinite algebraic genericity and spaceability for sequence spaces

“…We note that Papathanasiou in [6] proved that ℓ ∞ \ c 0 is maximal algebraically generic in ℓ ∞ , thus answering the question of whether Theorem 1.5 holds in the case X = ℓ ∞ . It is worth mentioning that algebraic genericity is often referred to as dense lineability in the literature.…”