The space Lp,λn (X) which is the space of all unbounded functions on the interval X = (0, 1] is defined. A special norm has been defined on the space  Lp,λn (X). Furthermore, the concept of the best multiplier approximation has been established. Additionally, new sets have been defined such as, the set of best multiplier approximation, a set for proximinal, the set for coproximinal , and a set for semi-Cochebyshev. All the previous sets are defined on Lp,λn (X). In the space Lp,λn (X)  some results of the best multiplier co-approximation of the unbounded functions defined on X = (0, 1] have been obtained. Some special sets in Lp,λn (X) have been studied. Several concepts have been built and employed to lead to the best multiplier co- approximation.