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 one 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
• James' theorem
• Mazur's lemma