A PROOF THAT EVERY BANACH SPACE IS SUBREFLEXIVE

A real or complex normed space is subreflexive if those functionals which attain their supremum on the unit sphere S of E are normdense in E*, i.e., if for each f in E* and each ∈>0 there exist g in E* and x in S such that |g(x)| = ∥g∥ and ∥f−g∥<∈. There exist incomplete normed spaces which are not subreflexive [1]¹ as well as incomplete spaces which are subreflexive (e.g., a dense subspace of a Hilbert space). It is evident that every reflexive Banach space is subreflexive. The theorem mentioned in the title will be proved for real Banach spaces; the result for complex spaces follows from this by considering the spaces over the real field and using the known isometry between complex functionals and the real functionals defined by their real parts…