一个被遗忘的Petrovczyovski定理：$(λ+)$ -内射空间不必是PDMATHAAB-内射--$λ\in (1,2]$情形
A forgotten theorem of Pełczyński: $(λ+)$-injective spaces need not be $λ$-injective -- the case $λ\in (1,2]$
摘要：
Isbell和Semadeni [Trans.Amer.Math.Soc.107（1963）]证明了每个无限维$1$ -内射Banach空间包含一个超平面，该超平面对于每个$\varepsilon > 0$是$(2+\varepsilon)$ -内射的，然而，它是\emph{not} $2$ -内射的，并在脚注中指出，Peczynski已经证明了每个$λ> 1$的存在性的$(λ+ \varepsilon)$ -内射空间（$\varepsilon > 0$）不是$λ$ -内射的。不幸的是，没有任何痕迹的证明Pejeczybroski的结果已被保存。本文通过构造$\ell_\infty$的一个适当的重赋范，建立了$λ\in (1,2]$的上述定理.这与Lindenstrauss [Mem. Amer. Math. Soc. 48（1964）]证明相反陈述的$λ= 1$案例形成鲜明对比（至少对于实标量）。
英文摘要：
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.
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