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 $f=h\circ\pi+x^{\ast}$, where $\pi :X\rightarrow X/Y$ 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.

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关于局部凸空间上凸函数的全局形状

在最近的论文 \cite{Aza:19} D Azagra 研究了定义在 Banach 空间 $X$ 上的连续凸函数的全局形状。更准确地说，当$X$可分时，证明对于每一个连续凸函数$f:X\rightarrow\mathbb{R}$都存在一个唯一的$X$闭线性子空间$Y$，一个连续函数$ h:X/Y\rightarrow\mathbb{R}$ 具有 $\lim_{t\rightarrow\infty}h(u+tv)=\infty$ 对于所有 $u,v\in X/Y$ 的属性， $v\neq0$ 和 $x^{\ast}\in X^{\ast}$ 使得 $f=h\circ\pi+x^{\ast}$，其中 $\pi :X\rightarrow X/Y$ 是自然投影。我们的目标是表征那些定义在具有上述表示的局部凸空间上的适当的下半连续凸函数。特别是，