论文标题
pełczyński的被遗忘的定理：$（λ+）$ - 注射空间不必是$λ$ - 注射器 - 案例$λ\ in（1,2] $
A forgotten theorem of Pełczyński: $(λ+)$-injective spaces need not be $λ$-injective -- the case $λ\in (1,2]$
论文作者
论文摘要
Isbell和Semadeni [Trans。阿米尔。数学。 Soc。 107 (1963)] proved that every infinite-dimensional $1$-injective Banach space contains a hyperplane that is $(2+\varepsilon)$-injective for every $\varepsilon > 0$, yet is is \emph{not} $2$-injective and remarked in a footnote that Pełczyński had proved for every $λ> 1$ the existence $（λ+ \ varepsilon）$ - 注射空间（$ \ varepsilon> 0 $），不是$λ$ - 注射器。不幸的是，尚未保留过Pełczyński结果的证据。在本文中，我们通过构建$ \ ell_ \ infty $的适当重置$ \ ell_ \ infty $的$λ\ in（1,2] $中的所述定理。这种对比（至少对于真实标量）与case $λ= 1 $ $λ= 1 $，lindenstrauss [mem。Amer。Math。Math。Soc。48（1964）]得到了相反的陈述。
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 $\varepsilon > 0$, yet is is \emph{not} $2$-injective and remarked in a footnote that Pełczyński had proved for every $λ> 1$ the existence of a $(λ+ \varepsilon)$-injective space ($\varepsilon > 0$) that is not $λ$-injective. Unfortunately, no trace of the proof of Pełczyński's result has been preserved. In the present paper, we establish the said theorem for $λ\in (1,2]$ by constructing an appropriate renorming of $\ell_\infty$. This contrasts (at least for real scalars) with the case $λ= 1$ for which Lindenstrauss [Mem. Amer. Math. Soc. 48 (1964)] proved the contrary statement.
