Goldstine theorem

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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 BX under the canonical imbedding into the closed unit ball B′′ of the bidual space X ′′ is weakly*-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 .

Proof[edit]

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

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 φX ′ such that φ|Y = 0, φ(x) ≥ 1 + δ and ||φ||X ′ = 1. Then φ ∈ span{φ1, ..., φn}, and therefore

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

which is a contradiction.

See also[edit]