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 BX 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 .

Proof

Lemma

For all , and , there exists an such that for all .

Proof of Lemma

By the surjectivity of

we can find with for .

Now let

Every element of z ∈ (x + Y) ∩ (1 + δ)B satisfies and , so it suffices to show that the intersection is nonempty.

Assume for contradiction 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} [1] and therefore

which is a contradiction.

Proof of Theorem

Fix , and . Examine the set

Let be the embedding defined by , where is the evaluation at map. Sets of the form form a base for the weak* topology,[2] so density follows if we can show for all such . The lemma above says that for all there exists an such that . Since , we have . We can scale to get . The goal is to show that for a sufficiently small , we have .

Directly checking, we have

.

Note that we can choose sufficiently large so that for .[3] Note as well that . If we choose so that , then we have that

Hence we get as desired.

See also

References

  1. Rudin, Walter. Functional Analysis (Second ed.). Lemma 3.9. pp. 63–64.CS1 maint: location (link)
  2. Rudin, Walter. Functional Analysis (Second ed.). Equation (3) and the remark after. p. 69.CS1 maint: location (link)
  3. Folland, Gerald. Real Analysis: Modern Techniques and Their Applications (Second ed.). Proposition 5.2. pp. 153–154.CS1 maint: location (link)
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