| On the basis of a finite measure space(Ω,Σ,μ)we define the left lim-it space Lp-0(μ,X)and the right limit space Lp+0(μ,X)for Bochner integrable space Lp(μ,X).By the means of locally convex topological vector space theo-rems,we obtain that Lp-0(μ,X)is a Local convex separated and both complete paranormed space,on the other hand,Lp+0(μ,X)is barrelled and borrological.Depending on the continuous embedding relations of Lp(μ,X),we receive a continuous embedding theorem for Lp(μ,X),Lp-0(μ,X)and Lp+0(μ,X).The dual of Bochner integrable space Lp(μ,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(μ,X).Then we receive that if X is reflexive,then Lp-0(μ,X),Lp+0(v,X)(1<p,q<∞),L∞-0(μ,X)and L1+0(μ,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(μ,X). |
