A real or complex normed space is subreflexive if those f unctionals which attain their supremum on the unit sphere 5 of E are normdense in £*, i.e., if for each / in £* and each €>0 there exist g in E* and x in 5 such that \g(x)\ =\\g\\ and ||/-g|| <€. There exist in complete normed spaces which are not subreflexive [l] 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 sub reflexive. 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 functional and the real functionals defined by their real parts.We first cite a lemma which states, roughly, that if the hyperplanes determined by two functionals / and g (of norm one) are nearly parallel, then one of ||/-g||, ||/+g|| must be small.
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