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, c₀, and its bi-dual space ℓ_∞.

Proof

Given x′′ ∈ B′′, an n-tuple (φ₁, ..., φ_n) of linearly independent elements of X ′ and a δ > 0 we shall find x in (1 + δ)B such that φ_i (x) = x′′(φ_i) for 1 ≤ i ≤ n.

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

$\begin{cases} \Phi : X \to \mathbf{C}^{n}, \\ x \mapsto \left (\varphi_1(x), \cdots, \varphi_n(x) \right ) \end{cases}$

Now let

$Y := \bigcap_i \ker \varphi_i = \ker \Phi.$

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{φ₁, ..., φ_n}, and therefore

$1+\delta \leq \varphi(x) = x''(\varphi) \leq \|\varphi\|_{X'} \left \|x'' \right \|_{X''} \leq 1,$

which is a contradiction.

See also

• Banach–Alaoglu theorem
• Bishop–Phelps theorem
• Eberlein–Šmulian theorem
• Mazur's lemma
• James' theorem