We prove the following main result: Let X be a normed linear space,fn ∈ X*\{θ},Hn = {x ∈ X: fn(x) = l},n = 0, 1,2,...Then w* - limfn = f0 iff H0 lim inf Hn and θ limsup Hn; when X is a reflexive Banach space, lim ||fn - f0|| = 0. If and only if θ w-limsup Hn Ho It simplifies the related results in [1].
In this paper we change the several important theorems in Calculus, about these theorems. angle of view to consider the connections among and give alternative proofs and new implications
