In functional analysis , a branch of mathematics, the Goldstine theorem , named after Herman Goldstine , is stated as follows:
Goldstine theorem. Let
X
{\displaystyle X}
be a Banach space , then the image of the closed unit ball
B
⊆
X
{\displaystyle B\subseteq X}
under the canonical embedding into the closed unit ball
B
′
′
{\displaystyle B^{\prime \prime }}
of the bidual space
X
′
′
{\displaystyle X^{\prime \prime }}
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
c
0
,
{\displaystyle c_{0},}
and its bi-dual space Lp space
ℓ
∞
.
{\displaystyle \ell ^{\infty }.}
For all
x
′
′
∈
B
′
′
,
{\displaystyle x^{\prime \prime }\in B^{\prime \prime },}
φ
1
,
…
,
φ
n
∈
X
′
{\displaystyle \varphi _{1},\ldots ,\varphi _{n}\in X^{\prime }}
and
δ
>
0
,
{\displaystyle \delta >0,}
there exists an
x
∈
(
1
+
δ
)
B
{\displaystyle x\in (1+\delta )B}
such that
φ
i
(
x
)
=
x
′
′
(
φ
i
)
{\displaystyle \varphi _{i}(x)=x^{\prime \prime }(\varphi _{i})}
for all
1
≤
i
≤
n
.
{\displaystyle 1\leq i\leq n.}
By the surjectivity of
{
Φ
:
X
→
C
n
,
x
↦
(
φ
1
(
x
)
,
⋯
,
φ
n
(
x
)
)
{\displaystyle {\begin{cases}\Phi :X\to \mathbb {C} ^{n},\\x\mapsto \left(\varphi _{1}(x),\cdots ,\varphi _{n}(x)\right)\end{cases}}}
it is possible to find
x
∈
X
{\displaystyle x\in X}
with
φ
i
(
x
)
=
x
′
′
(
φ
i
)
{\displaystyle \varphi _{i}(x)=x^{\prime \prime }(\varphi _{i})}
for
1
≤
i
≤
n
.
{\displaystyle 1\leq i\leq n.}
Now let
Y
:=
⋂
i
ker
φ
i
=
ker
Φ
.
{\displaystyle Y:=\bigcap _{i}\ker \varphi _{i}=\ker \Phi .}
Every element of
z
∈
(
x
+
Y
)
∩
(
1
+
δ
)
B
{\displaystyle z\in (x+Y)\cap (1+\delta )B}
satisfies
z
∈
(
1
+
δ
)
B
{\displaystyle z\in (1+\delta )B}
and
φ
i
(
z
)
=
φ
i
(
x
)
=
x
′
′
(
φ
i
)
,
{\displaystyle \varphi _{i}(z)=\varphi _{i}(x)=x^{\prime \prime }(\varphi _{i}),}
so it suffices to show that the intersection is nonempty.
Assume for contradiction that it is empty. Then
dist
(
x
,
Y
)
≥
1
+
δ
{\displaystyle \operatorname {dist} (x,Y)\geq 1+\delta }
and by the Hahn–Banach theorem there exists a linear form
φ
∈
X
′
{\displaystyle \varphi \in X^{\prime }}
such that
φ
|
Y
=
0
,
φ
(
x
)
≥
1
+
δ
{\displaystyle \varphi {\big \vert }_{Y}=0,\varphi (x)\geq 1+\delta }
and
‖
φ
‖
X
′
=
1.
{\displaystyle \|\varphi \|_{X^{\prime }}=1.}
Then
φ
∈
span
{
φ
1
,
…
,
φ
n
}
{\displaystyle \varphi \in \operatorname {span} \left\{\varphi _{1},\ldots ,\varphi _{n}\right\}}
[ 1] and therefore
1
+
δ
≤
φ
(
x
)
=
x
′
′
(
φ
)
≤
‖
φ
‖
X
′
‖
x
′
′
‖
X
′
′
≤
1
,
{\displaystyle 1+\delta \leq \varphi (x)=x^{\prime \prime }(\varphi )\leq \|\varphi \|_{X^{\prime }}\left\|x^{\prime \prime }\right\|_{X^{\prime \prime }}\leq 1,}
which is a contradiction.
Fix
x
′
′
∈
B
′
′
,
{\displaystyle x^{\prime \prime }\in B^{\prime \prime },}
φ
1
,
…
,
φ
n
∈
X
′
{\displaystyle \varphi _{1},\ldots ,\varphi _{n}\in X^{\prime }}
and
ϵ
>
0.
{\displaystyle \epsilon >0.}
Examine the set
U
:=
{
y
′
′
∈
X
′
′
:
|
(
x
′
′
−
y
′
′
)
(
φ
i
)
|
<
ϵ
,
1
≤
i
≤
n
}
.
{\displaystyle U:=\left\{y^{\prime \prime }\in X^{\prime \prime }:|(x^{\prime \prime }-y^{\prime \prime })(\varphi _{i})|<\epsilon ,1\leq i\leq n\right\}.}
Let
J
:
X
→
X
′
′
{\displaystyle J:X\rightarrow X^{\prime \prime }}
be the embedding defined by
J
(
x
)
=
Ev
x
,
{\displaystyle J(x)={\text{Ev}}_{x},}
where
Ev
x
(
φ
)
=
φ
(
x
)
{\displaystyle {\text{Ev}}_{x}(\varphi )=\varphi (x)}
is the evaluation at
x
{\displaystyle x}
map. Sets of the form
U
{\displaystyle U}
form a base for the weak* topology,[ 2] so density follows once it is shown
J
(
B
)
∩
U
≠
∅
{\displaystyle J(B)\cap U\neq \varnothing }
for all such
U
.
{\displaystyle U.}
The lemma above says that for any
δ
>
0
{\displaystyle \delta >0}
there exists a
x
∈
(
1
+
δ
)
B
{\displaystyle x\in (1+\delta )B}
such that
x
′
′
(
φ
i
)
=
φ
i
(
x
)
,
{\displaystyle x^{\prime \prime }(\varphi _{i})=\varphi _{i}(x),}
1
≤
i
≤
n
,
{\displaystyle 1\leq i\leq n,}
and in particular
Ev
x
∈
U
.
{\displaystyle {\text{Ev}}_{x}\in U.}
Since
J
(
B
)
⊂
B
′
′
,
{\displaystyle J(B)\subset B^{\prime \prime },}
we have
Ev
x
∈
(
1
+
δ
)
J
(
B
)
∩
U
.
{\displaystyle {\text{Ev}}_{x}\in (1+\delta )J(B)\cap U.}
We can scale to get
1
1
+
δ
Ev
x
∈
J
(
B
)
.
{\displaystyle {\frac {1}{1+\delta }}{\text{Ev}}_{x}\in J(B).}
The goal is to show that for a sufficiently small
δ
>
0
,
{\displaystyle \delta >0,}
we have
1
1
+
δ
Ev
x
∈
J
(
B
)
∩
U
.
{\displaystyle {\frac {1}{1+\delta }}{\text{Ev}}_{x}\in J(B)\cap U.}
Directly checking, one has
|
[
x
′
′
−
1
1
+
δ
Ev
x
]
(
φ
i
)
|
=
|
φ
i
(
x
)
−
1
1
+
δ
φ
i
(
x
)
|
=
δ
1
+
δ
|
φ
i
(
x
)
|
.
{\displaystyle \left|\left[x^{\prime \prime }-{\frac {1}{1+\delta }}{\text{Ev}}_{x}\right](\varphi _{i})\right|=\left|\varphi _{i}(x)-{\frac {1}{1+\delta }}\varphi _{i}(x)\right|={\frac {\delta }{1+\delta }}|\varphi _{i}(x)|.}
Note that one can choose
M
{\displaystyle M}
sufficiently large so that
‖
φ
i
‖
X
′
≤
M
{\displaystyle \|\varphi _{i}\|_{X^{\prime }}\leq M}
for
1
≤
i
≤
n
.
{\displaystyle 1\leq i\leq n.}
[ 3] Note as well that
‖
x
‖
X
≤
(
1
+
δ
)
.
{\displaystyle \|x\|_{X}\leq (1+\delta ).}
If one chooses
δ
{\displaystyle \delta }
so that
δ
M
<
ϵ
,
{\displaystyle \delta M<\epsilon ,}
then
δ
1
+
δ
|
φ
i
(
x
)
|
≤
δ
1
+
δ
‖
φ
i
‖
X
′
‖
x
‖
X
≤
δ
‖
φ
i
‖
X
′
≤
δ
M
<
ϵ
.
{\displaystyle {\frac {\delta }{1+\delta }}\left|\varphi _{i}(x)\right|\leq {\frac {\delta }{1+\delta }}\|\varphi _{i}\|_{X^{\prime }}\|x\|_{X}\leq \delta \|\varphi _{i}\|_{X^{\prime }}\leq \delta M<\epsilon .}
Hence one gets
1
1
+
δ
Ev
x
∈
J
(
B
)
∩
U
{\displaystyle {\frac {1}{1+\delta }}{\text{Ev}}_{x}\in J(B)\cap U}
as desired.
^ Rudin, Walter. Functional Analysis (Second ed.). Lemma 3.9. pp. 63–64. {{cite book }}
: CS1 maint: location (link )
^ Rudin, Walter. Functional Analysis (Second ed.). Equation (3) and the remark after. p. 69. {{cite book }}
: CS1 maint: location (link )
^ Folland, Gerald. Real Analysis: Modern Techniques and Their Applications (Second ed.). Proposition 5.2. pp. 153–154. {{cite book }}
: CS1 maint: location (link )
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