On the Generalized Real Convex Banach Spaces

Abstract

Given a normed linear space X, suppose its second dual X** exists so that a canonical injection J: X+X** also exists and is defined for each x ∈ X by J (x) = Øx where Øx: X**→ℝ is given by Øx (f)=˂f, x> for each f ∈ X* and <J (x),(f)>=˂f, x> for each f ∈ X**. Then, the mapping J is said to be embedded in X** and X is a reflexive Banach space in which the canonical embedding is onto. In this work, a general review of Kakutan’s, Helly’s, Goldstein’s theorem, and other propositions on the convex spaces was X-rayed before comprehensive results on uniformly convex spaces were studied, while the generalization of these results was discussed in section there as main result along the accompanying proofs to the result.