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

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

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 space and its bi-dual space [[Lp space]

Proof

Lemma

For all and there exists an such that for all

Proof of Lemma

By the surjectivity of

it is possible to find with for

Now let

Every element of satisfies and so it suffices to show that the intersection is nonempty.

Assume for contradiction that it is empty. Then and by the Hahn–Banach theorem there exists a linear form such that and Then [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 once it is shown for all such The lemma above says that for any there exists a such that and in particular 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)