The Left And Right Limit Spaces Both Their Dual Pairs For Bochner Integrable Space L~p(?,X)

On the basis of a finite measure space(?,?,?)we define the left lim-it space Lp-0(?,X)and the right limit space Lp+0(?,X)for Bochner integrable space Lp(?,X).By the means of locally convex topological vector space theo-rems,we obtain that Lp-0(?,X)is a Local convex separated and both complete paranormed space,on the other hand,Lp+0(?,X)is barrelled and borrological.Depending on the continuous embedding relations of Lp(?,X),we receive a continuous embedding theorem for Lp(?,X),Lp-0(?,X)and Lp+0(?,X).The dual of Bochner integrable space Lp(?,X)is depend on the property of Banach space X,therefore,we focus on the Banach space X which it's dual space has Radon-Nikodym property.Applying the Diestel's conclusions,we receive the dual spaces of the the left limit and right limit space of Lp(?,X).Then we receive that if X is reflexive,then Lp-0(?,X),Lp+0(v,X)(1<p,q<?),L?-0(?,X)and L1+0(?,X)is also reflexive.Finally we receive the dual pair of the left and right spaces,definition of polar topologies and some properties for Lp(?,X).