论文标题
在局部凸空间上凸的全局形状
On the global shape of convex functions on locally convex spaces
论文作者
论文摘要
在最近的论文中\ cite {aza：19} d Azagra研究了在Banach Space $ x $上定义的连续凸功能的全球形状。更准确地说，当$ x $可分开时，显示出每个连续的凸函数$ f：x \ rightarrow \ mathbb {r} $存在一个唯一的封闭的线性subpace $ y $ y $ x $的$ x $，连续函数$ h：x/y \ rightarrow \ rightarrow \ rightarrow \ mathbb {r} $ $ \ lim_ {t \ rightarrow \ infty}是自然投影。我们的目的是表征那些在具有上述表示的局部凸空间上定义的适当的下半连续凸功能。特别是，我们表明函数$ f $的连续性以及$ x $的完整性可以从Azagra定理的假设中删除。
In the recent paper \cite{Aza:19} D Azagra studies the global shape of continuous convex functions defined on a Banach space $X$. More precisely, when $X$ is separable, it is shown that for every continuous convex function $f:X\rightarrow\mathbb{R}$ there exist a unique closed linear subspace $Y$ of $X$, a continuous function $h:X/Y\rightarrow\mathbb{R}$ with the property that $\lim_{t\rightarrow\infty}h(u+tv)=\infty$ for all $u,v\in X/Y$, $v\neq0$, and $x^{\ast}\in X^{\ast}$ such that $f=h\circπ+x^{\ast}$, where $π:X\rightarrow X/Y$ is the natural projection. Our aim is to characterize those proper lower semi\-continuous convex functions defined on a locally convex space which have the above representation. In particular, we show that the continuity of the function $f$ and the completeness of $X$ can be removed from the hypothesis of Azagra's theorem.
