A renorming Characterisation of Banach Spaces containing ℓ 1 ( κ ) Avilés Redondo, Antonio Daniel Martínez Cervantes, Gonzalo Rueda Zoca, Abraham Ball-covering ℓ1(κ) Octahedral norm Renorming A result of G. Godefroy asserts that a Banach space X contains an isomorphic copy of ℓ 1 if and only if there is an equivalent norm | | | ⋅ | | | such that, for every finite-dimensional subspace Y of X and every ε > 0 , there exists x ∈ S X so that | | | y + r x | | | ≥ ( 1 − ε ) ( | | | y | | | + | r | ) for every y ∈ Y and every r ∈ R . In this paper we generalise this result to larger cardinals, showing that if κ is an uncountable cardinal, then a Banach space X contains a copy of ℓ 1 ( κ ) if and only if there is an equivalent norm | | | ⋅ | | | on X such that for every subspace Y of X with dens ( Y ) < κ there exists a norm-one vector x so that | | | y + r x | | | = | | | y | | | + | r | whenever y ∈ Y and r ∈ R . This result answers a question posed by S. Ciaci, J. Langemets, and A. Lissitsin, where the authors wonder whether the above statement holds for infinite successor cardinals. We also show that, in the countable case, the result of Godefroy cannot be improved to take ε = 0 . 2023-10-03T08:58:06Z 2023-10-03T08:58:06Z 2023 info:eu-repo/semantics/article Antonio Avilés. Gonzalo Martínez-Cervantes. Abraham Rueda Zoca. A renorming Characterisation of Banach Spaces containing ℓ 1 ( κ ) ." Publ. Mat. 67 (2) 601 - 609, 2023. https://doi.org/10.5565/PUBLMAT6722305 https://hdl.handle.net/10481/84800 10.5565/PUBLMAT6722305 eng http://creativecommons.org/licenses/by/4.0/ info:eu-repo/semantics/openAccess Atribución 4.0 Internacional Universitat Autònoma de Barcelona