In functional analysis, a branch of mathematics, the Goldstine theorem, named after Herman Goldstine, asserts that the image of the closed unit ball of a Banach space under the canonical imbedding into the closed unit ball of the bidual space is weakly*-dense.
Given an , a tuple of linearly independent elements of and a we shall find an such that for every .
If the requirement is dropped, the existence of such an follows from the surjectivity of
Let now . Every element of has the required property, so that it suffices to show that the latter set is not empty.
Assume that it is empty. Then and by the Hahn-Banach theorem there exists a linear form such that , and . Then and therefore
which is a contradiction.