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In order to study the conditions for bounded closed and convex sets to have a unique completion inreal Banach spaces，known results in this direction are summarized. Based on this，a sufficient condition as well as some necessary and sufficient conditions for bounded closed and convex sets to have a unique completion areprovided. The notion of ( K，u) -completeness is extended，and the relation of this notion to the uniqueness of completion in real Banach spaces is discussed.