This study discussed the integral of Dunford and compact linear operator on space of Dunford integral function. For each f which is Dunford integral on [a,b] is defined as an operator DL by DL (x *) = x * f, for each x * ∈ X * This study resulted that the operator DL is both a continuous linear operator and weakly compact operators. Then, it was defined as the adjoint of the operator D L * by D L * ( h ) ( x * ) = ∫ a b h D L ( x * ) each h ∈ (L 1)* The adjoint operator D L * is continuous and weakly compact linear operators.

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