A forgotten theorem of Pełczyński: $(\lambda +)$-injective spaces need not be $\lambda $-injective—the case $\lambda \in (1,2]$
Isbell and Semadeni [Trans. Amer. Math. Soc. 107 (1963)] proved that every infinite-dimensional $1$-injective Banach space contains a hyperplane that is $(2+\varepsilon )$-injective for every $\varepsilon \gt 0$, yet is not $2$-injective, and remarked in a footnote that Pełczyński had proved for every $\lambda \gt 1$ the existence of a $(\lambda + \varepsilon )$-injective space ($\varepsilon \gt 0$) that is not $\lambda $-injective. Unfortunately, no trace of the proof of Pełczyński’s result has been preserved. In the present paper, we establish that result for $\lambda \in (1,2]$ by constructing an appropriate renorming of $\ell _\infty $. This contrasts (at least for real scalars) with the case $\lambda = 1$ for which Lindenstrauss [Mem. Amer. Math. Soc. 48 (1964)] proved the contrary statement.