In
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defi ...
, a branch of mathematics, the Goldstine theorem, named after
Herman Goldstine
Herman Heine Goldstine (September 13, 1913 – June 16, 2004) was a mathematician and computer scientist, who worked as the director of the IAS machine at Princeton University's Institute for Advanced Study and helped to develop ENIAC, th ...
, is stated as follows:
:Goldstine theorem. Let
be a
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
, 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
In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the ra ...
.
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
In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numb ...
and its bi-dual space
Lp space
In mathematics, the spaces are function spaces defined using a natural generalization of the -norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although according to the Bourb ...
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
The Hahn–Banach theorem is a central tool in functional analysis.
It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear f ...
there exists a linear form
such that
and
Then
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, 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, one has
Note that one can choose
sufficiently large so that
for
Note as well that
If one chooses
so that
then
Hence one gets
as desired.
See also
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References
*
{{Functional Analysis
Banach spaces
Theorems in functional analysis
de:Schwach-*-Topologie#Eigenschaften