# Goldstine theorem

In functional analysis, a branch of mathematics, the **Goldstine theorem**, named after Herman Goldstine, is stated as follows:

**Goldstine theorem.**Let X be a Banach space, then the image of the closed unit ball*B*⊂*X*under the canonical embedding into the closed unit ball*B*′′ of the bidual space*X*′′ is weak*-dense.

The conclusion of the theorem is not true for the norm topology, which can be seen by considering the Banach space of real sequences that converge to zero, *c*_{0}, and its bi-dual space ℓ_{∞}.

## Proof

Given *x*′′ ∈ *B*′′, an n-tuple (*φ*_{1}, ..., *φ _{n}*) of linearly independent elements of

*X*′ and a

*δ*> 0 one shall find x in (1 +

*δ*)

*B*such that

*φ*(

_{i}*x*) =

*x*′′(

*φ*) for 1 ≤

_{i}*i*≤

*n*.

If the requirement ||*x*|| ≤ 1 + *δ* is dropped, the existence of such an x follows from the surjectivity of

Now let

Every element of (*x* + *Y*) ∩ (1 + *δ*)*B* has the required property, so that it suffices to show that the latter set is not empty.

Assume that it is empty. Then dist(*x*, *Y*) ≥ 1 + *δ* and by the Hahn–Banach theorem there exists a linear form *φ* ∈ *X* ′ such that *φ*|_{Y} = 0, *φ*(*x*) ≥ 1 + *δ* and ||*φ*||_{X ′} = 1. Then *φ* ∈ span{*φ*_{1}, ..., *φ _{n}*}, and therefore

which is a contradiction.