Using a lifting of £∞(μ,X) ([5],[6]), we construct a lifting ρxof the seminormed vector space £∞(μ,X) of measurable, essentially boundedX-valued functions. We show that in a certain sense such a lifting always exists. If μ is Lebesgue measure on (0, 1) we show that ρxexists as map from £∞((O, 1),X) → £∞,((0, l),X) if and only if X is reflexive. In general the lifted function takes its values inX**. Therefore we investigate the question, when f ε £∞(μ,X) is strictly liftable in the sense that the lifted function is a map with values even inX.