This paper investigates the existence of complex support functional of a convex function in complex linear spaces and the complex numerical ranges of the complex support functionals on a point.

采用子空间中支撑泛函延拓的方法，构造出在复线性空间任意点上的复支撑泛函；确定在同一支撑点上复支撑泛函的数值域，从而得到复支撑泛函具有唯一性的充分必要条件。

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