论文标题
Hahn-Banach扩展和相关规范的独特性 - 双空间中的预测$ 1 $
Uniqueness of Hahn-Banach extension and related norm-$1$ projections in dual spaces
论文作者
论文摘要
在本文中，我们研究了两种属性。属性-U $和属性-U $ SU $的子空间$ y $的Banach空间，这对应于$ y^*$中每个线性功能的Hahn-Banach扩展的唯一性，此外，此关联还形成了Normate-1的线性运算符，从$ Y^**$ to $ x^*$。事实证明，在$ x，y，z $上的某些几何假设下，这些属性相对于注射量张量产品是稳定的。 $ y $具有$ u $（$ su $）的$ z $，并且仅当$ x \ otimes_ \ e^\ vee y $具有property- $ u $（$ su $）中的$ x \ otimes_ \ otimes_ \ e^\ vee z $。我们证明，当$ x^*$具有radon-nikod $ \急性{y} $ m属性，$ 1 <p <\ infty $，$ l_p（μ，y，y）$具有property-$ u $ $ u $（property-$ su $）$ l_p（μ，x）$，并且只有$ y $ so y in $ x $。我们表明，如果$ z \ subseteq y \ subseteq x $，其中$ y $具有property- $ u $（$ su $），则$ x $，然后$ y/z $具有$ x/z $的property- $ u $（$ su $）。另一方面，如果$ y/z $具有property- $ su $ in $ x/z $，而$ z $和$ z（\ subseteq y）$是$ x $的m-dideal，则$ y $ $ y $具有$ x $ in $ x $中的$ su $。据观察，当且仅在任何两个子空间$ y，z $中，$ x $ in $ x $，$ y+z $，只要$ y+y y+y y+y+z $关闭，$ y+z $ in $ x $ in $ x $ in $ x $ in $ x $ in $ x $ in $ y+y+y+y+y+y+y+z $，我们表征了$ C_0 $中的所有超平面，其中$ $ $ $。
In this paper we study two properties viz. property-$U$ and property-$SU$ of a subspace $Y$ of a Banach space which correspond to the uniqueness of the Hahn-Banach extension of each linear functional in $Y^*$ and in addition to that this association forms a linear operator of norm-1 from $Y^*$ to $X^*$. It is proved that, under certain geometric assumptions on $X, Y, Z$ these properties are stable with respect to the injective tensor product; $Y$ has property-$U$ ($SU$) in $Z$ if and only if $X\otimes_\e^\vee Y$ has property-$U$ ($SU$) in $X\otimes_\e^\vee Z$. We prove that when $X^*$ has the Radon-Nikod$\acute{y}$m Property for $1<p< \infty$, $L_p(μ, Y)$ has property-$U$ (property-$SU$) in $L_p(μ, X)$ if and only if $Y$ is so in $X$. We show that if $Z\subseteq Y\subseteq X$, where $Y$ has property-$U$ ($SU$) in $X$ then $Y/Z$ has property-$U$ ($SU$) in $X/Z$. On the other hand $Y$ has property-$SU$ in $X$ if $Y/Z$ has property-$SU$ in $X/Z$ and $Z (\subseteq Y)$ is an M-ideal in $X$. It is observed that a smooth Banach space of dimension $>3$ is a Hilbert space if and only if for any two subspaces $Y, Z$ with property-$SU$ in $X$, $Y+Z$ has property-$SU$ in $X$ whenever $Y+Z$ is closed. We characterize all hyperplanes in $c_0$ which have property-$SU$.
