Reflexive space

A reflexive space is a Banach space X for which the natural embedding into its double dual X** is surjective — equivalently, a space that is isometrically isomorphic to its second dual. Reflexivity is the infinite-dimensional analogue of finite-dimensionality: in a reflexive space, the closed unit ball is weakly compact, bounded sequences have weakly convergent subsequences, and every nonempty closed convex set has an element of minimal norm. These properties fail catastrophically in non-reflexive spaces such as L¹ and L^∞. The distinction between reflexive and non-reflexive spaces is not a mere taxonomic curiosity; it marks the boundary between spaces where optimization problems are well-posed and spaces where they dissolve into pathological counterexamples. The geometry of dual spaces is what determines reflexivity, and reflexivity is what makes infinite-dimensional analysis feel almost finite.