We present the following unbounded version for James's theorem on weak compactness in Banach spaces: let C be a closed, convex but not necessarily bounded subset in the Banach space E, and Λ be a non-void and $\tau (E^{⁎},E)$-open subset of $E^{⁎}$; i.e. Mackey open in the dual space, such that$\sup \{z^{⁎}(c):c\in C\}<+\infty \text{ whenever }{z}^{⁎}\in \mathrm{\Lambda }.$ If C is not $\sigma (E^{⁎⁎},E^{⁎})$-closed in $E^{⁎⁎}$ there is a linear form $z^{⁎}\in \mathrm{\Lambda }$ such that the $\sup \{z^{⁎}(c):c\in C\}$ is not attained.

As a main application we have the following: if $f:\mathbb{E}\to \mathbb{R}\cup \{\infty \}$ is a proper function such that the range of the subdifferential $\partial f(E)$ contains a nonvoid open subset for the Mackey topology on the dual space $(E^{⁎},\tau (E^{⁎},E))$, then for each set $c\in \mathbb{R}$ the sublevel set $f^{-1}((-\infty ,c]$ is relatively weakly compact. If in addition the function f has a domain with non-empty norm interior, the Banach space E must be reflexive.

Straightforward applications to robust representation of risk measures and monotone operators are also derived.

对于在Banach空间中的弱紧致性的James定理，我们给出以下无界版本：令C是Banach空间E中的封闭，凸但不一定有界的子集，而Λ是非无效且$\tau （E^{⁎}，E）$-打开子集 $E^{⁎}$; 即Mackey在双重空间中打开，这样$\mathrm{SUP}\{{ž}^{⁎}（C）：C\in C\}<+\infty \text{ 每当 }{ž}^{⁎}\in \mathrm{\Lambda }。$如果C不是$\sigma （E^{⁎⁎}，E^{⁎}）$-封闭于 $E^{⁎⁎}$ 有线性形式 ${ž}^{⁎}\in \mathrm{\Lambda }$ 这样 $\mathrm{SUP}\{{ž}^{⁎}（C）：C\in C\}$ 达不到。

作为主要应用程序，我们具有以下内容： $F：\mathbb{E}\to \mathbb{[R}\cup \{\infty \}$ 是适当的函数，使得微分的范围 $\partial F（E）$ 在双空间上包含Mackey拓扑的非空开放子集 $（E^{⁎}，\tau （E^{⁎}，E））$，然后每组 $C\in \mathbb{[R}$ 子级集 $F^{-\mathrm{1个}}（（-\infty ，C]$相对较弱的紧凑性。另外，如果函数f具有非空范数内部的域，则Banach空间E必须是自反的。

还导出了对风险度量和单调运算符的可靠表示的简单应用。